↑

Properties

Label 1-1617-1617.1025-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.934 + 0.354i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (0.251 − 0.967i)2-s + (−0.873 − 0.486i)4-s + (−0.337 + 0.941i)5-s + (−0.691 + 0.722i)8-s + (0.826 + 0.563i)10-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (0.599 − 0.800i)17-s + (0.669 + 0.743i)19-s + (0.753 − 0.657i)20-s + (−0.955 − 0.294i)23-s + (−0.772 − 0.635i)25-s + (0.0149 + 0.999i)26-s + (−0.995 + 0.0896i)29-s + (0.104 − 0.994i)31-s + (0.955 − 0.294i)32-s + ⋯
L(s)  = 1  + (0.251 − 0.967i)2-s + (−0.873 − 0.486i)4-s + (−0.337 + 0.941i)5-s + (−0.691 + 0.722i)8-s + (0.826 + 0.563i)10-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (0.599 − 0.800i)17-s + (0.669 + 0.743i)19-s + (0.753 − 0.657i)20-s + (−0.955 − 0.294i)23-s + (−0.772 − 0.635i)25-s + (0.0149 + 0.999i)26-s + (−0.995 + 0.0896i)29-s + (0.104 − 0.994i)31-s + (0.955 − 0.294i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.934 + 0.354i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.934 + 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135977957 + 0.2082975957i\)
\(L(\frac12)\) \(\approx\) \(1.135977957 + 0.2082975957i\)
\(L(1)\) \(\approx\) \(0.8596435963 - 0.2923115238i\)
\(L(1)\) \(\approx\) \(0.8596435963 - 0.2923115238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.251 - 0.967i)T \)
5 \( 1 + (-0.337 + 0.941i)T \)
13 \( 1 + (-0.963 + 0.266i)T \)
17 \( 1 + (0.599 - 0.800i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
41 \( 1 + (0.691 - 0.722i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.998 - 0.0598i)T \)
53 \( 1 + (-0.992 + 0.119i)T \)
59 \( 1 + (-0.280 + 0.959i)T \)
61 \( 1 + (0.992 + 0.119i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.753 - 0.657i)T \)
73 \( 1 + (0.998 + 0.0598i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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

−20.19421300559818899240372786969, −19.48255125478713382971749726840, −18.666588184250481052378228803738, −17.527282957986893123201427526603, −17.30958759963099504998150932847, −16.32670616986914586201539591361, −15.87614292425448405370056895342, −15.03908937390018621832454314739, −14.356923955462426403355756905212, −13.49303544480902035681792881120, −12.70219078351654362040933935002, −12.25101783544282475187767253467, −11.3232594101117409727238867522, −9.93449439958445740846202969370, −9.41932105996339924484527379735, −8.48742947116345272421845287763, −7.810359797083043691005989354936, −7.23522138214656701432650454481, −6.06637872158221688825846988472, −5.37472197875316746123886751871, −4.648918594347298815196762844124, −3.88765034810185577018642064033, −2.881805240546222434851830308999, −1.37433936812589090438578147396, −0.29096152589672394164053768081, 0.68844598061139506225312974306, 2.08147215778762978799256924313, 2.64564852246165917196304263334, 3.651810740814838613012408105, 4.24742601477782554746239822447, 5.40519789822458456085914504959, 6.075263483942265860661514659397, 7.38599260410768947449334512866, 7.81225734754999560468259592683, 9.19721274420953653775583357646, 9.75705553875190143906414850232, 10.50420333005744121044607113503, 11.25559227090909481785302392283, 12.01164387804382758051265917705, 12.44705155642077208037244172855, 13.659453157202784516529529980698, 14.26800456758769810144197985880, 14.72714642679010848375690399107, 15.6845824498509477561413251264, 16.64088495004225706055292366416, 17.624341109640710628166648913, 18.37958900671206199945210965434, 18.86096846829616927580966306165, 19.540281008465217459355872409540, 20.3443948200559851936330541443

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