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Properties

Label 2-3460-3460.807-c0-0-0
Degree $2$
Conductor $3460$
Sign $0.423 + 0.906i$
Analytic cond. $1.72676$
Root an. cond. $1.31406$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.288 − 0.957i)2-s + (−0.833 − 0.551i)4-s + (0.424 + 0.905i)5-s + (−0.768 + 0.639i)8-s + (0.997 + 0.0729i)9-s + (0.989 − 0.145i)10-s + (−0.289 − 0.538i)13-s + (0.391 + 0.920i)16-s + (−0.939 − 1.21i)17-s + (0.357 − 0.934i)18-s + (0.145 − 0.989i)20-s + (−0.639 + 0.768i)25-s + (−0.598 + 0.121i)26-s + (1.65 − 0.563i)29-s + (0.994 − 0.109i)32-s + ⋯
L(s)  = 1  + (0.288 − 0.957i)2-s + (−0.833 − 0.551i)4-s + (0.424 + 0.905i)5-s + (−0.768 + 0.639i)8-s + (0.997 + 0.0729i)9-s + (0.989 − 0.145i)10-s + (−0.289 − 0.538i)13-s + (0.391 + 0.920i)16-s + (−0.939 − 1.21i)17-s + (0.357 − 0.934i)18-s + (0.145 − 0.989i)20-s + (−0.639 + 0.768i)25-s + (−0.598 + 0.121i)26-s + (1.65 − 0.563i)29-s + (0.994 − 0.109i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3460\)    =    \(2^{2} \cdot 5 \cdot 173\)
Sign: $0.423 + 0.906i$
Analytic conductor: \(1.72676\)
Root analytic conductor: \(1.31406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3460} (807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3460,\ (\ :0),\ 0.423 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.515504913\)
\(L(\frac12)\) \(\approx\) \(1.515504913\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.288 + 0.957i)T \)
5 \( 1 + (-0.424 - 0.905i)T \)
173 \( 1 + (0.920 - 0.391i)T \)
good3 \( 1 + (-0.997 - 0.0729i)T^{2} \)
7 \( 1 + (-0.520 + 0.853i)T^{2} \)
11 \( 1 + (-0.768 + 0.639i)T^{2} \)
13 \( 1 + (0.289 + 0.538i)T + (-0.551 + 0.833i)T^{2} \)
17 \( 1 + (0.939 + 1.21i)T + (-0.252 + 0.967i)T^{2} \)
19 \( 1 + (-0.424 + 0.905i)T^{2} \)
23 \( 1 + (0.768 + 0.639i)T^{2} \)
29 \( 1 + (-1.65 + 0.563i)T + (0.791 - 0.611i)T^{2} \)
31 \( 1 + (-0.997 + 0.0729i)T^{2} \)
37 \( 1 + (-1.68 + 1.07i)T + (0.424 - 0.905i)T^{2} \)
41 \( 1 + (-0.886 - 1.57i)T + (-0.520 + 0.853i)T^{2} \)
43 \( 1 + (0.920 + 0.391i)T^{2} \)
47 \( 1 + (-0.288 - 0.957i)T^{2} \)
53 \( 1 + (-1.18 + 0.309i)T + (0.872 - 0.489i)T^{2} \)
59 \( 1 + (-0.889 + 0.457i)T^{2} \)
61 \( 1 + (-0.210 - 1.63i)T + (-0.967 + 0.252i)T^{2} \)
67 \( 1 + (0.0729 - 0.997i)T^{2} \)
71 \( 1 + (-0.611 - 0.791i)T^{2} \)
73 \( 1 + (0.536 - 0.779i)T + (-0.357 - 0.934i)T^{2} \)
79 \( 1 + (0.288 - 0.957i)T^{2} \)
83 \( 1 + (0.999 - 0.0365i)T^{2} \)
89 \( 1 + (0.0855 - 0.777i)T + (-0.976 - 0.217i)T^{2} \)
97 \( 1 + (-1.22 + 1.18i)T + (0.0365 - 0.999i)T^{2} \)
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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

−8.902855987419823673018449273059, −7.87932513358387077411710767122, −7.08482155025605226523583980499, −6.35106456316224437136808508913, −5.49953437091798759381297001784, −4.58666422651876393602613781889, −3.98522630349097011071107980235, −2.69006385476599833129659457132, −2.47589822965382121510837782034, −1.03926485725121020164223753719, 1.16478370381214275850956809299, 2.42437934477491254283578811922, 3.91758211406822462785011517172, 4.43218814168116430550057648408, 5.03260857824723279757313030914, 6.07786399212807080339823646262, 6.52710119834826665828225690766, 7.36685386076842905682394148152, 8.170134145420426344354207750892, 8.786930017599826965038218733315

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