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Properties

Label 2-680-136.101-c1-0-20
Degree $2$
Conductor $680$
Sign $-0.257 - 0.966i$
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 + 0.303i)2-s − 0.228·3-s + (1.81 + 0.838i)4-s − 5-s + (−0.316 − 0.0694i)6-s + 3.99i·7-s + (2.25 + 1.70i)8-s − 2.94·9-s + (−1.38 − 0.303i)10-s − 3.75·11-s + (−0.415 − 0.191i)12-s + 3.12i·13-s + (−1.21 + 5.51i)14-s + 0.228·15-s + (2.59 + 3.04i)16-s + (3.25 + 2.53i)17-s + ⋯
L(s)  = 1  + (0.976 + 0.214i)2-s − 0.132·3-s + (0.907 + 0.419i)4-s − 0.447·5-s + (−0.129 − 0.0283i)6-s + 1.50i·7-s + (0.796 + 0.604i)8-s − 0.982·9-s + (−0.436 − 0.0959i)10-s − 1.13·11-s + (−0.119 − 0.0553i)12-s + 0.865i·13-s + (−0.323 + 1.47i)14-s + 0.0590·15-s + (0.648 + 0.760i)16-s + (0.788 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.257 - 0.966i$
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: \((2,\ 680,\ (\ :1/2),\ -0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24197 + 1.61652i\)
\(L(\frac12)\) \(\approx\) \(1.24197 + 1.61652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.303i)T \)
5 \( 1 + T \)
17 \( 1 + (-3.25 - 2.53i)T \)
good3 \( 1 + 0.228T + 3T^{2} \)
7 \( 1 - 3.99iT - 7T^{2} \)
11 \( 1 + 3.75T + 11T^{2} \)
13 \( 1 - 3.12iT - 13T^{2} \)
19 \( 1 + 1.96iT - 19T^{2} \)
23 \( 1 + 0.990iT - 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 + 2.33iT - 31T^{2} \)
37 \( 1 - 6.42T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 - 3.10iT - 43T^{2} \)
47 \( 1 + 0.730T + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 - 14.8iT - 59T^{2} \)
61 \( 1 - 8.99T + 61T^{2} \)
67 \( 1 - 7.33iT - 67T^{2} \)
71 \( 1 - 0.713iT - 71T^{2} \)
73 \( 1 - 7.10iT - 73T^{2} \)
79 \( 1 - 8.64iT - 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 18.7iT - 97T^{2} \)
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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

−11.20083450132914658948743923874, −10.01341556543399704436314788924, −8.639971566293649109435275970674, −8.216778411632741343413531275082, −7.05777381267310745540574544860, −5.91043830064054175459850295967, −5.48082074991995830617119006263, −4.42674358017971705910944356431, −3.03957376536855214873713353495, −2.32295851581813319485586249730, 0.78588242380956968744545365721, 2.80648253634725720992555385879, 3.55982761799826968376230125104, 4.74245638585743837415719616548, 5.48964344207475382069748359729, 6.56396200204971326319036860887, 7.67859960749878350501874452400, 8.004310382319274713485050431743, 9.794620598931472626295954182167, 10.56478009492723011520622720288

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