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Properties

Label 2-680-680.509-c1-0-89
Degree $2$
Conductor $680$
Sign $-0.358 + 0.933i$
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  + (1.20 − 0.744i)2-s − 1.24i·3-s + (0.890 − 1.79i)4-s + (1.91 − 1.16i)5-s + (−0.928 − 1.49i)6-s + 1.17·7-s + (−0.262 − 2.81i)8-s + 1.44·9-s + (1.43 − 2.81i)10-s − 6.18·11-s + (−2.23 − 1.11i)12-s + 3.23·13-s + (1.41 − 0.878i)14-s + (−1.44 − 2.38i)15-s + (−2.41 − 3.19i)16-s + (2.20 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s − 0.719i·3-s + (0.445 − 0.895i)4-s + (0.854 − 0.519i)5-s + (−0.378 − 0.611i)6-s + 0.445·7-s + (−0.0928 − 0.995i)8-s + 0.482·9-s + (0.453 − 0.891i)10-s − 1.86·11-s + (−0.644 − 0.320i)12-s + 0.897·13-s + (0.379 − 0.234i)14-s + (−0.373 − 0.615i)15-s + (−0.603 − 0.797i)16-s + (0.535 + 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.358 + 0.933i$
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: \((2,\ 680,\ (\ :1/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62302 - 2.36142i\)
\(L(\frac12)\) \(\approx\) \(1.62302 - 2.36142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.744i)T \)
5 \( 1 + (-1.91 + 1.16i)T \)
17 \( 1 + (-2.20 - 3.48i)T \)
good3 \( 1 + 1.24iT - 3T^{2} \)
7 \( 1 - 1.17T + 7T^{2} \)
11 \( 1 + 6.18T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
19 \( 1 - 6.25iT - 19T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + 5.33T + 29T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + 3.17iT - 37T^{2} \)
41 \( 1 + 0.353iT - 41T^{2} \)
43 \( 1 - 7.73T + 43T^{2} \)
47 \( 1 - 3.04iT - 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 12.0iT - 59T^{2} \)
61 \( 1 - 6.46T + 61T^{2} \)
67 \( 1 + 8.64T + 67T^{2} \)
71 \( 1 - 5.66iT - 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 + 1.11iT - 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 1.72T + 89T^{2} \)
97 \( 1 + 3.78T + 97T^{2} \)
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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

−10.44022026375154474762911031400, −9.686736936040525213710333156385, −8.315467348440869276285907713409, −7.61045244612597410018212921751, −6.30045577210757681679545584378, −5.64801298625242918157624151261, −4.83830416324821667001204563763, −3.53537485584305262231487697696, −2.09731880314056642516948718049, −1.36269003233897287720649559606, 2.29161556699309870749359489317, 3.24543609271839534078874871982, 4.55184213482457256237400013482, 5.26684873889002796314218365696, 6.02817446907460486467834480208, 7.23667338707108446599275691001, 7.84744023619958237270559452689, 9.097520002012090967806950776907, 9.970394727994906768850563829538, 10.93361924529398181502170358129

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