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Properties

Label 2-680-680.509-c1-0-53
Degree $2$
Conductor $680$
Sign $0.524 + 0.851i$
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 + (3.16 − 0.0281i)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 + (0.999 − 0.00890i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.524 + 0.851i$
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: \((2,\ 680,\ (\ :1/2),\ 0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868916 - 0.485094i\)
\(L(\frac12)\) \(\approx\) \(0.868916 - 0.485094i\)
\(L(\frac{3}{2})\) 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

−9.990151937911821417345766152016, −9.456726601931855942356817016717, −8.363297712180816836384145138119, −7.76215749926616193755496699376, −7.06331938697392603873913756845, −6.23591652969460226064328946643, −4.96749126655656649619368915084, −3.90628283713269089037368820644, −1.88644672246399433589457550051, −0.841209613192447588981569189235, 1.34565153097403020266644389047, 3.02320645079981355502903375745, 3.97625418218408799676678548710, 4.69996124463059303400751593867, 6.71424796129609505893466538917, 7.09501128932980268686100394055, 8.296876687784380096180708474296, 8.973005658772375997508318907565, 9.900427538550623567958953337713, 10.50771968848524534665639824996

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