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Properties

Label 2-680-85.4-c1-0-13
Degree $2$
Conductor $680$
Sign $0.375 - 0.926i$
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  + (1.68 + 1.68i)3-s + (2.22 + 0.179i)5-s + (−0.196 + 0.196i)7-s + 2.68i·9-s + (−0.480 − 0.480i)11-s + 5.44i·13-s + (3.45 + 4.05i)15-s + (−3.39 + 2.33i)17-s − 2.29i·19-s − 0.662·21-s + (5.23 − 5.23i)23-s + (4.93 + 0.798i)25-s + (0.532 − 0.532i)27-s + (−1.54 + 1.54i)29-s + (−0.282 + 0.282i)31-s + ⋯
L(s)  = 1  + (0.973 + 0.973i)3-s + (0.996 + 0.0801i)5-s + (−0.0742 + 0.0742i)7-s + 0.894i·9-s + (−0.144 − 0.144i)11-s + 1.50i·13-s + (0.892 + 1.04i)15-s + (−0.823 + 0.566i)17-s − 0.527i·19-s − 0.144·21-s + (1.09 − 1.09i)23-s + (0.987 + 0.159i)25-s + (0.102 − 0.102i)27-s + (−0.286 + 0.286i)29-s + (−0.0508 + 0.0508i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.375 - 0.926i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.375 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95468 + 1.31650i\)
\(L(\frac12)\) \(\approx\) \(1.95468 + 1.31650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.22 - 0.179i)T \)
17 \( 1 + (3.39 - 2.33i)T \)
good3 \( 1 + (-1.68 - 1.68i)T + 3iT^{2} \)
7 \( 1 + (0.196 - 0.196i)T - 7iT^{2} \)
11 \( 1 + (0.480 + 0.480i)T + 11iT^{2} \)
13 \( 1 - 5.44iT - 13T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + (-5.23 + 5.23i)T - 23iT^{2} \)
29 \( 1 + (1.54 - 1.54i)T - 29iT^{2} \)
31 \( 1 + (0.282 - 0.282i)T - 31iT^{2} \)
37 \( 1 + (5.09 + 5.09i)T + 37iT^{2} \)
41 \( 1 + (-0.0550 - 0.0550i)T + 41iT^{2} \)
43 \( 1 - 6.98T + 43T^{2} \)
47 \( 1 + 3.42iT - 47T^{2} \)
53 \( 1 + 4.63T + 53T^{2} \)
59 \( 1 + 2.84iT - 59T^{2} \)
61 \( 1 + (10.3 + 10.3i)T + 61iT^{2} \)
67 \( 1 - 0.574iT - 67T^{2} \)
71 \( 1 + (-2.98 + 2.98i)T - 71iT^{2} \)
73 \( 1 + (-10.8 - 10.8i)T + 73iT^{2} \)
79 \( 1 + (9.61 + 9.61i)T + 79iT^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (3.25 + 3.25i)T + 97iT^{2} \)
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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

−10.60393865739812315512414674224, −9.483895327683342603391145852323, −9.116907259130053268549293098097, −8.514747505218796999225579170342, −7.03421744839410429082739748558, −6.25807841317014010934220656218, −4.94062367338223491991185710246, −4.16307437167747895810425579176, −2.93798201310391086898787784685, −1.97857610385703748956100889532, 1.27610387804497979092686504626, 2.45269091173925945583703568475, 3.26078156715609659776667534870, 5.00051747492920024044573544029, 5.91398634429413347932970050208, 6.99629414633322438962577453576, 7.67749228136520489660592934673, 8.583819923098347106600342051857, 9.318075288147779788507304804950, 10.19426399670056871302853556556

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