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Properties

Label 2-3460-3460.563-c0-0-0
Degree $2$
Conductor $3460$
Sign $0.676 + 0.736i$
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  + (0.946 − 0.322i)2-s + (0.791 − 0.611i)4-s + (−0.288 − 0.957i)5-s + (0.551 − 0.833i)8-s + (0.457 + 0.889i)9-s + (−0.581 − 0.813i)10-s + (1.79 + 0.880i)13-s + (0.252 − 0.967i)16-s + (0.468 + 0.999i)17-s + (0.719 + 0.694i)18-s + (−0.813 − 0.581i)20-s + (−0.833 + 0.551i)25-s + (1.98 + 0.254i)26-s + (0.0789 + 0.354i)29-s + (−0.0729 − 0.997i)32-s + ⋯
L(s)  = 1  + (0.946 − 0.322i)2-s + (0.791 − 0.611i)4-s + (−0.288 − 0.957i)5-s + (0.551 − 0.833i)8-s + (0.457 + 0.889i)9-s + (−0.581 − 0.813i)10-s + (1.79 + 0.880i)13-s + (0.252 − 0.967i)16-s + (0.468 + 0.999i)17-s + (0.719 + 0.694i)18-s + (−0.813 − 0.581i)20-s + (−0.833 + 0.551i)25-s + (1.98 + 0.254i)26-s + (0.0789 + 0.354i)29-s + (−0.0729 − 0.997i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3460\)    =    \(2^{2} \cdot 5 \cdot 173\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(1.72676\)
Root analytic conductor: \(1.31406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3460} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3460,\ (\ :0),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.475107396\)
\(L(\frac12)\) \(\approx\) \(2.475107396\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.946 + 0.322i)T \)
5 \( 1 + (0.288 + 0.957i)T \)
173 \( 1 + (0.967 + 0.252i)T \)
good3 \( 1 + (-0.457 - 0.889i)T^{2} \)
7 \( 1 + (0.934 + 0.357i)T^{2} \)
11 \( 1 + (0.551 - 0.833i)T^{2} \)
13 \( 1 + (-1.79 - 0.880i)T + (0.611 + 0.791i)T^{2} \)
17 \( 1 + (-0.468 - 0.999i)T + (-0.639 + 0.768i)T^{2} \)
19 \( 1 + (0.288 - 0.957i)T^{2} \)
23 \( 1 + (-0.551 - 0.833i)T^{2} \)
29 \( 1 + (-0.0789 - 0.354i)T + (-0.905 + 0.424i)T^{2} \)
31 \( 1 + (-0.457 + 0.889i)T^{2} \)
37 \( 1 + (1.14 + 1.54i)T + (-0.288 + 0.957i)T^{2} \)
41 \( 1 + (1.88 + 0.347i)T + (0.934 + 0.357i)T^{2} \)
43 \( 1 + (0.967 - 0.252i)T^{2} \)
47 \( 1 + (-0.946 - 0.322i)T^{2} \)
53 \( 1 + (0.652 - 0.543i)T + (0.181 - 0.983i)T^{2} \)
59 \( 1 + (-0.667 + 0.744i)T^{2} \)
61 \( 1 + (-1.74 + 0.630i)T + (0.768 - 0.639i)T^{2} \)
67 \( 1 + (-0.889 + 0.457i)T^{2} \)
71 \( 1 + (-0.424 - 0.905i)T^{2} \)
73 \( 1 + (0.671 + 1.66i)T + (-0.719 + 0.694i)T^{2} \)
79 \( 1 + (0.946 - 0.322i)T^{2} \)
83 \( 1 + (0.853 - 0.520i)T^{2} \)
89 \( 1 + (-0.504 - 0.0369i)T + (0.989 + 0.145i)T^{2} \)
97 \( 1 + (0.349 + 0.623i)T + (-0.520 + 0.853i)T^{2} \)
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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

−8.589265776392404861722480196337, −7.988242902450107066585716952208, −7.03105200471763833653227708615, −6.26473468101460047323674971890, −5.45172217288627163170830053707, −4.83899113936614652086264757158, −3.91369343788531451795384211322, −3.55438296993252332852223674422, −1.90200633527032305660151747865, −1.42699577977634100069636612276, 1.43101640848733219175294404884, 2.95037641347338562117574044529, 3.37284353750933614890053042625, 4.05879700714223335669191042132, 5.15329126903249779808513639309, 5.94150477934218083624233307015, 6.69179225186999682248231499476, 7.00097782985464010556164893470, 8.098056099229729332314748897066, 8.472998424456585705781785543130

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