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Properties

Label 2-680-680.509-c1-0-21
Degree $2$
Conductor $680$
Sign $-0.636 + 0.771i$
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  + (0.483 + 1.32i)2-s + 2.44i·3-s + (−1.53 + 1.28i)4-s + (−0.380 + 2.20i)5-s + (−3.24 + 1.18i)6-s + 2.45·7-s + (−2.44 − 1.41i)8-s − 2.97·9-s + (−3.11 + 0.560i)10-s − 2.47·11-s + (−3.14 − 3.74i)12-s − 2.44·13-s + (1.18 + 3.25i)14-s + (−5.38 − 0.929i)15-s + (0.694 − 3.93i)16-s + (4.12 − 0.0180i)17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + 1.41i·3-s + (−0.766 + 0.642i)4-s + (−0.170 + 0.985i)5-s + (−1.32 + 0.482i)6-s + 0.926·7-s + (−0.866 − 0.499i)8-s − 0.991·9-s + (−0.984 + 0.177i)10-s − 0.746·11-s + (−0.907 − 1.08i)12-s − 0.679·13-s + (0.316 + 0.870i)14-s + (−1.39 − 0.239i)15-s + (0.173 − 0.984i)16-s + (0.999 − 0.00436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.636 + 0.771i$
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: \((2,\ 680,\ (\ :1/2),\ -0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622082 - 1.32017i\)
\(L(\frac12)\) \(\approx\) \(0.622082 - 1.32017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 - 1.32i)T \)
5 \( 1 + (0.380 - 2.20i)T \)
17 \( 1 + (-4.12 + 0.0180i)T \)
good3 \( 1 - 2.44iT - 3T^{2} \)
7 \( 1 - 2.45T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
19 \( 1 - 1.36iT - 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 - 6.21iT - 31T^{2} \)
37 \( 1 - 2.43iT - 37T^{2} \)
41 \( 1 + 7.78iT - 41T^{2} \)
43 \( 1 - 5.04T + 43T^{2} \)
47 \( 1 - 0.293iT - 47T^{2} \)
53 \( 1 + 9.16T + 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + 5.56T + 61T^{2} \)
67 \( 1 - 8.90T + 67T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + 4.79T + 73T^{2} \)
79 \( 1 - 8.84iT - 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 3.27T + 97T^{2} \)
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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

−10.74667383405262849081365621842, −10.20328903641171276659324064595, −9.369452618862332301751065725456, −8.233269294273846017056376095213, −7.64072471942285893691445910935, −6.58817691929414922528940332455, −5.32030227322855715415979749779, −4.86375475225402161961564446088, −3.77825159395330021030612310235, −2.86896306677190051723288802706, 0.72763326613925328357969881112, 1.72504650181059018054243617226, 2.80125935457172385429967634544, 4.48067932960082773206103513502, 5.17822897381845691505595645994, 6.16461972093132565805362924842, 7.70327357245367052174311684220, 7.998032957890357305260840489032, 9.054880029702167663903825350190, 10.03539043317173654313302675595

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