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Properties

Label 2-680-17.8-c1-0-11
Degree $2$
Conductor $680$
Sign $0.770 - 0.637i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.805 + 1.94i)3-s + (0.923 + 0.382i)5-s + (4.50 − 1.86i)7-s + (−1.01 − 1.01i)9-s + (0.738 + 1.78i)11-s − 4.49i·13-s + (−1.48 + 1.48i)15-s + (2.20 − 3.48i)17-s + (5.38 − 5.38i)19-s + 10.2i·21-s + (1.69 + 4.09i)23-s + (0.707 + 0.707i)25-s + (−3.04 + 1.26i)27-s + (−8.52 − 3.53i)29-s + (1.27 − 3.08i)31-s + ⋯
L(s)  = 1  + (−0.465 + 1.12i)3-s + (0.413 + 0.171i)5-s + (1.70 − 0.704i)7-s + (−0.338 − 0.338i)9-s + (0.222 + 0.537i)11-s − 1.24i·13-s + (−0.384 + 0.384i)15-s + (0.534 − 0.845i)17-s + (1.23 − 1.23i)19-s + 2.23i·21-s + (0.353 + 0.854i)23-s + (0.141 + 0.141i)25-s + (−0.586 + 0.242i)27-s + (−1.58 − 0.655i)29-s + (0.229 − 0.553i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63271 + 0.587356i\)
\(L(\frac12)\) \(\approx\) \(1.63271 + 0.587356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-2.20 + 3.48i)T \)
good3 \( 1 + (0.805 - 1.94i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-4.50 + 1.86i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.738 - 1.78i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.49iT - 13T^{2} \)
19 \( 1 + (-5.38 + 5.38i)T - 19iT^{2} \)
23 \( 1 + (-1.69 - 4.09i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (8.52 + 3.53i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.27 + 3.08i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.96 - 9.58i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.90 - 0.788i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-5.75 - 5.75i)T + 43iT^{2} \)
47 \( 1 - 6.13iT - 47T^{2} \)
53 \( 1 + (4.96 - 4.96i)T - 53iT^{2} \)
59 \( 1 + (-5.21 - 5.21i)T + 59iT^{2} \)
61 \( 1 + (6.49 - 2.68i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + (1.11 - 2.68i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.42 + 2.24i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.64 + 8.79i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.0745 + 0.0745i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + (-8.24 - 3.41i)T + (68.5 + 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58329198472142283455203270790, −9.845872495020033869571235401040, −9.178283933932624776035792389971, −7.70125196879183714550482416111, −7.41262136512894163835443401503, −5.66412503747784295807163684322, −5.03453782644562148099059848197, −4.39702324734528365503276777381, −3.05296894757918272879551483395, −1.30112076569385980402286057162, 1.43135245821557676538338122741, 1.94357071458924296670622794844, 3.88886970904000291386860669339, 5.31784976734970039051840874875, 5.77099880244339864593825346132, 6.89845922918635621586821628445, 7.74322799047147305219912553938, 8.547869937247655971593654399730, 9.323596441340248627579736614006, 10.68332006046362268815933789733

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