Properties

Label 1-675-675.23-r0-0-0
Degree 11
Conductor 675675
Sign 0.009300.999i0.00930 - 0.999i
Analytic cond. 3.134683.13468
Root an. cond. 3.134683.13468
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯
L(s)  = 1  + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯

Functional equation

Λ(s)=(675s/2ΓR(s)L(s)=((0.009300.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(675s/2ΓR(s)L(s)=((0.009300.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 675675    =    33523^{3} \cdot 5^{2}
Sign: 0.009300.999i0.00930 - 0.999i
Analytic conductor: 3.134683.13468
Root analytic conductor: 3.134683.13468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ675(23,)\chi_{675} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 675, (0: ), 0.009300.999i)(1,\ 675,\ (0:\ ),\ 0.00930 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.12824286490.1270546478i0.1282428649 - 0.1270546478i
L(12)L(\frac12) \approx 0.12824286490.1270546478i0.1282428649 - 0.1270546478i
L(1)L(1) \approx 0.7639118757+0.3577900787i0.7639118757 + 0.3577900787i
L(1)L(1) \approx 0.7639118757+0.3577900787i0.7639118757 + 0.3577900787i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
good2 1+(0.529+0.848i)T 1 + (0.529 + 0.848i)T
7 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
11 1+(0.03480.999i)T 1 + (-0.0348 - 0.999i)T
13 1+(0.529+0.848i)T 1 + (-0.529 + 0.848i)T
17 1+(0.406+0.913i)T 1 + (-0.406 + 0.913i)T
19 1+(0.104+0.994i)T 1 + (0.104 + 0.994i)T
23 1+(0.1390.990i)T 1 + (0.139 - 0.990i)T
29 1+(0.2410.970i)T 1 + (-0.241 - 0.970i)T
31 1+(0.7190.694i)T 1 + (-0.719 - 0.694i)T
37 1+(0.7430.669i)T 1 + (-0.743 - 0.669i)T
41 1+(0.8480.529i)T 1 + (-0.848 - 0.529i)T
43 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
47 1+(0.6940.719i)T 1 + (-0.694 - 0.719i)T
53 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
59 1+(0.03480.999i)T 1 + (0.0348 - 0.999i)T
61 1+(0.8820.469i)T 1 + (-0.882 - 0.469i)T
67 1+(0.970+0.241i)T 1 + (0.970 + 0.241i)T
71 1+(0.1040.994i)T 1 + (0.104 - 0.994i)T
73 1+(0.743+0.669i)T 1 + (-0.743 + 0.669i)T
79 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
83 1+(0.0697+0.997i)T 1 + (-0.0697 + 0.997i)T
89 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
97 1+(0.2750.961i)T 1 + (0.275 - 0.961i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.74235045015576027597638505814, −22.1961791988889169392937787219, −21.46603570782534809398694960260, −20.16051195001202459708565213678, −20.04681996510013441574707174632, −19.1220021330213394481690307579, −18.09250436021077649983838868492, −17.54789744877373118478234315394, −16.12494827006930737101009492149, −15.361926640019587946760445794357, −14.67090657124199754268914422463, −13.473651260816980243704904790369, −12.952930317184261688257366959352, −12.188555842110710557352696920519, −11.330512536092546058219135768283, −10.28223287591958839252349156353, −9.61621434952439398926204546893, −8.927717833829796014241440302021, −7.34587758393836083553910886810, −6.55551453841262875441658126685, −5.28891936562399101986530640491, −4.73150751217133362594985706116, −3.32740400975208251125932181457, −2.79192722375557864904222945263, −1.543112721931446707372746981677, 0.069380423234969512169798230553, 2.21169230323186325970743718374, 3.50840911398879826674975774246, 4.0575185163800822600828036838, 5.363035699901945329455984928396, 6.23860449184586505378287987515, 6.82743888261567445532060500523, 7.94629472383583606733294190692, 8.78718899778069848368506881163, 9.66332172561171784846692857502, 10.78318206617136458273028983072, 11.97724352860337499397966672669, 12.687229187205579565977679528639, 13.546213479600863397036108986057, 14.215259562823025714033333256585, 15.13353361203868027259839696755, 16.00464206559009499893786539098, 16.76061096036969763952195826713, 17.13037046019835190280991915737, 18.626152350934431098742864020423, 19.00052940697207612066005996086, 20.19465387751861112107043019337, 21.21594220934585685286740737566, 21.95369404168799654951420409181, 22.55790008203230475629940897768

Graph of the ZZ-function along the critical line