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Properties

Label 2-680-136.101-c1-0-36
Degree $2$
Conductor $680$
Sign $-0.0787 + 0.996i$
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  + (−1.26 − 0.634i)2-s − 1.91·3-s + (1.19 + 1.60i)4-s + 5-s + (2.42 + 1.21i)6-s − 3.32i·7-s + (−0.495 − 2.78i)8-s + 0.669·9-s + (−1.26 − 0.634i)10-s + 5.61·11-s + (−2.29 − 3.07i)12-s + 3.09i·13-s + (−2.11 + 4.20i)14-s − 1.91·15-s + (−1.14 + 3.83i)16-s + (−3.99 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)2-s − 1.10·3-s + (0.597 + 0.801i)4-s + 0.447·5-s + (0.988 + 0.495i)6-s − 1.25i·7-s + (−0.175 − 0.984i)8-s + 0.223·9-s + (−0.399 − 0.200i)10-s + 1.69·11-s + (−0.661 − 0.886i)12-s + 0.859i·13-s + (−0.564 + 1.12i)14-s − 0.494·15-s + (−0.285 + 0.958i)16-s + (−0.967 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.0787 + 0.996i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.0787 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452489 - 0.489651i\)
\(L(\frac12)\) \(\approx\) \(0.452489 - 0.489651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.634i)T \)
5 \( 1 - T \)
17 \( 1 + (3.99 + 1.03i)T \)
good3 \( 1 + 1.91T + 3T^{2} \)
7 \( 1 + 3.32iT - 7T^{2} \)
11 \( 1 - 5.61T + 11T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
19 \( 1 + 0.142iT - 19T^{2} \)
23 \( 1 - 1.97iT - 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 + 5.08iT - 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + 4.91iT - 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 5.17T + 47T^{2} \)
53 \( 1 + 0.833iT - 53T^{2} \)
59 \( 1 - 1.78iT - 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + 8.46iT - 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 4.67iT - 97T^{2} \)
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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

−10.49577497180971865804620370323, −9.378937972182728053551982691713, −8.937922791786652934663777694996, −7.47046501102064299159642505567, −6.70476524520373892866163098445, −6.18847257561381673507596903899, −4.55469275843706285000746362867, −3.70970671700577828286173396012, −1.86706676705967231879003233620, −0.63099220559603828253003853721, 1.23687063837242052084817905884, 2.67864962308141947107547995694, 4.70885634804337172193444082733, 5.75189335372985736608634479244, 6.20393245035209434584302507115, 6.91321251474079697748382276403, 8.408841918749865313567867083751, 8.929517453096976596259331197427, 9.764632756310330784891288554434, 10.71114628284561749276757839502

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