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Properties

Label 2-680-680.509-c1-0-49
Degree $2$
Conductor $680$
Sign $-0.480 + 0.876i$
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  + (−0.909 − 1.08i)2-s + 2.68i·3-s + (−0.347 + 1.96i)4-s + (−2.10 + 0.747i)5-s + (2.90 − 2.43i)6-s − 0.590·7-s + (2.44 − 1.41i)8-s − 4.18·9-s + (2.72 + 1.60i)10-s − 5.24·11-s + (−5.28 − 0.931i)12-s + 2.44·13-s + (0.536 + 0.639i)14-s + (−2.00 − 5.65i)15-s + (−3.75 − 1.36i)16-s + (−4.03 + 0.839i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + 1.54i·3-s + (−0.173 + 0.984i)4-s + (−0.942 + 0.334i)5-s + (1.18 − 0.994i)6-s − 0.223·7-s + (0.866 − 0.500i)8-s − 1.39·9-s + (0.861 + 0.507i)10-s − 1.58·11-s + (−1.52 − 0.268i)12-s + 0.679·13-s + (0.143 + 0.170i)14-s + (−0.517 − 1.45i)15-s + (−0.939 − 0.342i)16-s + (−0.979 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.480 + 0.876i$
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: \((2,\ 680,\ (\ :1/2),\ -0.480 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0239194 - 0.0403891i\)
\(L(\frac12)\) \(\approx\) \(0.0239194 - 0.0403891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 1.08i)T \)
5 \( 1 + (2.10 - 0.747i)T \)
17 \( 1 + (4.03 - 0.839i)T \)
good3 \( 1 - 2.68iT - 3T^{2} \)
7 \( 1 + 0.590T + 7T^{2} \)
11 \( 1 + 5.24T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
19 \( 1 + 2.57iT - 19T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 - 4.11iT - 31T^{2} \)
37 \( 1 + 8.73iT - 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 - 6.08T + 43T^{2} \)
47 \( 1 - 5.98iT - 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 7.40iT - 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 - 0.239iT - 71T^{2} \)
73 \( 1 - 5.97T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 6.83T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−10.53430375776388764742764303872, −9.334530593117817478802721401196, −8.836710617812926882267058695390, −7.901509058401424407246716581643, −6.94518392405056591330596269364, −5.22420104784823117609838449093, −4.36550530117132349164867187270, −3.47743455710401055703744180761, −2.68297717120987182377621829329, −0.03302195321993768631864271517, 1.29262033082805879187084617482, 2.80242647162562642560425558795, 4.64746627728707471027379052916, 5.70246993659599300292312960425, 6.69287412423295294600195874565, 7.34225962715366013850021196014, 8.129249818667940974645126332946, 8.481232211149669335187849416895, 9.681737271393275886976805360437, 10.99378673107873728963168422363

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