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Properties

Label 2-680-680.509-c1-0-78
Degree $2$
Conductor $680$
Sign $0.998 + 0.0523i$
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  + (1.29 + 0.564i)2-s − 2.10i·3-s + (1.36 + 1.46i)4-s + (2.12 + 0.687i)5-s + (1.19 − 2.73i)6-s + 2.16·7-s + (0.937 + 2.66i)8-s − 1.45·9-s + (2.36 + 2.09i)10-s + 1.85·11-s + (3.09 − 2.87i)12-s − 6.11·13-s + (2.80 + 1.22i)14-s + (1.45 − 4.48i)15-s + (−0.291 + 3.98i)16-s + (−0.319 + 4.11i)17-s + ⋯
L(s)  = 1  + (0.916 + 0.399i)2-s − 1.21i·3-s + (0.680 + 0.732i)4-s + (0.951 + 0.307i)5-s + (0.486 − 1.11i)6-s + 0.818·7-s + (0.331 + 0.943i)8-s − 0.483·9-s + (0.749 + 0.662i)10-s + 0.558·11-s + (0.892 − 0.829i)12-s − 1.69·13-s + (0.750 + 0.327i)14-s + (0.374 − 1.15i)15-s + (−0.0729 + 0.997i)16-s + (−0.0775 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.998 + 0.0523i$
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: \((2,\ 680,\ (\ :1/2),\ 0.998 + 0.0523i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.16334 - 0.0828750i\)
\(L(\frac12)\) \(\approx\) \(3.16334 - 0.0828750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.564i)T \)
5 \( 1 + (-2.12 - 0.687i)T \)
17 \( 1 + (0.319 - 4.11i)T \)
good3 \( 1 + 2.10iT - 3T^{2} \)
7 \( 1 - 2.16T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 + 6.11T + 13T^{2} \)
19 \( 1 + 2.40iT - 19T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 - 0.731iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 4.69iT - 41T^{2} \)
43 \( 1 - 5.90T + 43T^{2} \)
47 \( 1 + 3.82iT - 47T^{2} \)
53 \( 1 + 6.98T + 53T^{2} \)
59 \( 1 + 1.35iT - 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 4.82T + 73T^{2} \)
79 \( 1 - 0.112iT - 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−10.73060412040080037160217521769, −9.604187910686589787074386476836, −8.386177291436415825819636928982, −7.47003059529765136832523565914, −6.95566672336260745368485501795, −6.04361234067325111532993807535, −5.25389137914150318965890036024, −4.08711252203904387142331736412, −2.38882727788431111557695625107, −1.83672015636358544912312851341, 1.71059935405558758107251854109, 2.88829602383822284125793571707, 4.29050420125444964370505303483, 4.84529398810602678445526140829, 5.51804472842992798650030866176, 6.66032258134931113875828325452, 7.85244941124611287201575917188, 9.384746990837810709629599253675, 9.722337458759626521684126743642, 10.40080284508347909896941999845

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