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Properties

Label 2-680-680.509-c1-0-37
Degree $2$
Conductor $680$
Sign $-0.790 - 0.612i$
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  + (−0.909 + 1.08i)2-s + 2.68i·3-s + (−0.347 − 1.96i)4-s + (2.10 + 0.747i)5-s + (−2.90 − 2.43i)6-s + 0.590·7-s + (2.44 + 1.41i)8-s − 4.18·9-s + (−2.72 + 1.60i)10-s + 5.24·11-s + (5.28 − 0.931i)12-s + 2.44·13-s + (−0.536 + 0.639i)14-s + (−2.00 + 5.65i)15-s + (−3.75 + 1.36i)16-s + (4.03 − 0.839i)17-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + 1.54i·3-s + (−0.173 − 0.984i)4-s + (0.942 + 0.334i)5-s + (−1.18 − 0.994i)6-s + 0.223·7-s + (0.866 + 0.500i)8-s − 1.39·9-s + (−0.861 + 0.507i)10-s + 1.58·11-s + (1.52 − 0.268i)12-s + 0.679·13-s + (−0.143 + 0.170i)14-s + (−0.517 + 1.45i)15-s + (−0.939 + 0.342i)16-s + (0.979 − 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464092 + 1.35684i\)
\(L(\frac12)\) \(\approx\) \(0.464092 + 1.35684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 1.08i)T \)
5 \( 1 + (-2.10 - 0.747i)T \)
17 \( 1 + (-4.03 + 0.839i)T \)
good3 \( 1 - 2.68iT - 3T^{2} \)
7 \( 1 - 0.590T + 7T^{2} \)
11 \( 1 - 5.24T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
19 \( 1 - 2.57iT - 19T^{2} \)
23 \( 1 + 7.58T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 4.11iT - 31T^{2} \)
37 \( 1 + 8.73iT - 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 - 6.08T + 43T^{2} \)
47 \( 1 + 5.98iT - 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 + 6.60T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 - 0.239iT - 71T^{2} \)
73 \( 1 + 5.97T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 6.83T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−10.38861813390281173768845441866, −9.928522778162720320329551378168, −9.154319938983780514622761452329, −8.637103754356445259068081410500, −7.32927329281265038940304296835, −6.10198271552593675243691511079, −5.70110174544547783659589252179, −4.49428077213521966540209825730, −3.55170778374056652362730436197, −1.62842253375359474751341242755, 1.17706382138462477985310584381, 1.65435295485133619354675965351, 2.96482206889641107776153337110, 4.42001244046732315670841364459, 6.13832322066717815185773598115, 6.53024160442094951204037465686, 7.80704902990973470292908962837, 8.348825328251334562766900541192, 9.330494103621363309104939348720, 9.976619374517014779288098105419

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