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Properties

Label 2-5780-17.16-c1-0-39
Degree $2$
Conductor $5780$
Sign $0.410 + 0.911i$
Analytic cond. $46.1535$
Root an. cond. $6.79363$
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  − 2.30i·3-s + i·5-s + 1.53i·7-s − 2.30·9-s + 2.44i·11-s − 6.18·13-s + 2.30·15-s + 0.345·19-s + 3.52·21-s − 9.04i·23-s − 25-s − 1.60i·27-s + 5.05i·29-s + 2.71i·31-s + 5.63·33-s + ⋯
L(s)  = 1  − 1.32i·3-s + 0.447i·5-s + 0.579i·7-s − 0.767·9-s + 0.738i·11-s − 1.71·13-s + 0.594·15-s + 0.0793·19-s + 0.769·21-s − 1.88i·23-s − 0.200·25-s − 0.308i·27-s + 0.938i·29-s + 0.486i·31-s + 0.981·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5780\)    =    \(2^{2} \cdot 5 \cdot 17^{2}\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(46.1535\)
Root analytic conductor: \(6.79363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5780} (5201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5780,\ (\ :1/2),\ 0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513903208\)
\(L(\frac12)\) \(\approx\) \(1.513903208\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 + 2.30iT - 3T^{2} \)
7 \( 1 - 1.53iT - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + 6.18T + 13T^{2} \)
19 \( 1 - 0.345T + 19T^{2} \)
23 \( 1 + 9.04iT - 23T^{2} \)
29 \( 1 - 5.05iT - 29T^{2} \)
31 \( 1 - 2.71iT - 31T^{2} \)
37 \( 1 - 4.08iT - 37T^{2} \)
41 \( 1 - 1.14iT - 41T^{2} \)
43 \( 1 + 0.730T + 43T^{2} \)
47 \( 1 - 0.594T + 47T^{2} \)
53 \( 1 - 9.59T + 53T^{2} \)
59 \( 1 + 6.74T + 59T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 1.08iT - 71T^{2} \)
73 \( 1 + 9.34iT - 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 6.97iT - 97T^{2} \)
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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

−7.77975553480604102207728928917, −7.19478664656751994390089730872, −6.74065131060740363997320951865, −6.11868287843021566638521971557, −5.06833929298112410562496119840, −4.55193722397575006284503931455, −3.16374752074263549960604820073, −2.35000204418516994607181415005, −1.94174754753309377127823620623, −0.54872758305278421388028890360, 0.73182691918898281799742053767, 2.17352207488376703431927761323, 3.20568799675767873080319311155, 3.99593778532281431890572818747, 4.45462429009808348939356083208, 5.42893500431320514140006235586, 5.62883532779518287794389279656, 7.01025622796444784344242201226, 7.53316308928049655043086178483, 8.309193080517299212689390768079

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