↑

Properties

Label 1-1100-1100.147-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.345 + 0.938i$
Analytic cond. $5.10837$
Root an. cond. $5.10837$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)3-s + (0.587 + 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)13-s + (0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (0.587 + 0.809i)27-s + (0.809 + 0.587i)29-s − 31-s + (0.951 + 0.309i)37-s + (−0.809 + 0.587i)39-s + 41-s − i·43-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (0.587 + 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)13-s + (0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (0.587 + 0.809i)27-s + (0.809 + 0.587i)29-s − 31-s + (0.951 + 0.309i)37-s + (−0.809 + 0.587i)39-s + 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.870186372 + 1.303741830i\)
\(L(\frac12)\) \(\approx\) \(1.870186372 + 1.303741830i\)
\(L(1)\) \(\approx\) \(1.481467737 + 0.4491618958i\)
\(L(1)\) \(\approx\) \(1.481467737 + 0.4491618958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 - T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.193898938442598893100918283046, −20.411069762051026980095963364989, −19.59021731788383212171364001374, −19.307516166867078029334471219675, −18.07176592066064324394479832286, −17.51238693934011497098062758523, −16.71534120347443965020923907076, −15.56303111204651985480541259235, −14.7928843359806449640252268658, −14.39205755325836272666717846080, −13.273569948705258795377651938983, −12.92841159364492396289726488422, −11.864765955591656911283977533079, −10.77257011534770595545796186412, −10.11964799726857002627954215125, −9.179046038152878078117878782906, −8.25338157986088022893072314515, −7.63466890185388156795298433701, −6.970073735017564371208244004986, −5.79119000688863291954191700526, −4.63207230376362485106261649485, −3.835443261593927562620636354931, −2.89146003005570836238238367140, −1.89327256370139770283109288227, −0.871122530059217216193246553763, 1.47333511940722220166060670559, 2.39117234278609198223717797918, 3.10866573668438099687511854133, 4.39738727379867918606258042539, 4.90093745573138189761656339823, 6.11062891337539437700119714349, 7.22058492303676467699050665541, 7.98849300533597604945450091964, 8.84059533255101524387598240363, 9.37746977578971048545024345420, 10.31277958551475891302538385076, 11.24466454580763662285175877865, 12.19534897306570581489602664505, 12.88340052184918161608855031003, 13.98703740547130598792305685529, 14.68404043010983698710722187774, 14.94652081366098947595476603229, 16.212795709108632275645248225293, 16.58419090235058487086126057088, 17.90733717965936231569240404888, 18.61020866191771620007410613949, 19.21144733166124152068459826027, 20.04394264719372763361438477943, 20.93493532563713222468079021966, 21.38406384213517392354271305336

Graph of the $Z$-function along the critical line