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Properties

Label 2-680-136.101-c1-0-59
Degree $2$
Conductor $680$
Sign $0.973 - 0.227i$
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  + (1.38 + 0.299i)2-s + 2.30·3-s + (1.82 + 0.826i)4-s − 5-s + (3.19 + 0.690i)6-s − 2.59i·7-s + (2.27 + 1.68i)8-s + 2.33·9-s + (−1.38 − 0.299i)10-s + 0.888·11-s + (4.20 + 1.90i)12-s + 1.64i·13-s + (0.774 − 3.58i)14-s − 2.30·15-s + (2.63 + 3.01i)16-s + (−2.66 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.977 + 0.211i)2-s + 1.33·3-s + (0.910 + 0.413i)4-s − 0.447·5-s + (1.30 + 0.281i)6-s − 0.979i·7-s + (0.802 + 0.596i)8-s + 0.778·9-s + (−0.437 − 0.0945i)10-s + 0.267·11-s + (1.21 + 0.551i)12-s + 0.454i·13-s + (0.207 − 0.957i)14-s − 0.596·15-s + (0.658 + 0.752i)16-s + (−0.646 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.973 - 0.227i$
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: \((2,\ 680,\ (\ :1/2),\ 0.973 - 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.80917 + 0.438194i\)
\(L(\frac12)\) \(\approx\) \(3.80917 + 0.438194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.299i)T \)
5 \( 1 + T \)
17 \( 1 + (2.66 - 3.14i)T \)
good3 \( 1 - 2.30T + 3T^{2} \)
7 \( 1 + 2.59iT - 7T^{2} \)
11 \( 1 - 0.888T + 11T^{2} \)
13 \( 1 - 1.64iT - 13T^{2} \)
19 \( 1 - 2.15iT - 19T^{2} \)
23 \( 1 + 8.19iT - 23T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 + 3.93iT - 31T^{2} \)
37 \( 1 + 1.45T + 37T^{2} \)
41 \( 1 - 1.15iT - 41T^{2} \)
43 \( 1 - 8.22iT - 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 4.95iT - 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 7.11iT - 67T^{2} \)
71 \( 1 - 4.08iT - 71T^{2} \)
73 \( 1 - 7.46iT - 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 1.74iT - 83T^{2} \)
89 \( 1 - 7.73T + 89T^{2} \)
97 \( 1 + 4.92iT - 97T^{2} \)
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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

−10.68050944608011104146608958959, −9.608451422711528864848245180550, −8.420428756674566759047564376249, −7.956347384488501343133042546231, −6.99602543092572037719439812138, −6.22412065052691728276221448660, −4.54561714472550146015193825145, −4.00503846726478315978461972090, −3.09248702164999011807558651588, −1.88545420217640448863463294237, 1.91089190204498932971000235310, 2.93659832687046913906969714627, 3.59500314651155465578679703135, 4.83284632978293419763875958356, 5.77108878387462797020551397874, 7.03517006268802272807046573623, 7.75598966966495825594931887368, 8.842064495386940636002285732959, 9.384362311305564769535205438208, 10.55968712085913162288348376734

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