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Properties

Label 2-1340-1340.1299-c0-0-0
Degree $2$
Conductor $1340$
Sign $-0.983 + 0.179i$
Analytic cond. $0.668747$
Root an. cond. $0.817769$
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  + (0.580 + 0.814i)2-s + (−0.550 + 0.353i)3-s + (−0.327 + 0.945i)4-s + (−0.142 + 0.989i)5-s + (−0.607 − 0.243i)6-s + (−0.0947 + 0.00904i)7-s + (−0.959 + 0.281i)8-s + (−0.237 + 0.520i)9-s + (−0.888 + 0.458i)10-s + (−0.154 − 0.635i)12-s + (−0.0623 − 0.0719i)14-s + (−0.271 − 0.595i)15-s + (−0.786 − 0.618i)16-s + (−0.561 + 0.108i)18-s + (−0.888 − 0.458i)20-s + (0.0489 − 0.0384i)21-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (−0.550 + 0.353i)3-s + (−0.327 + 0.945i)4-s + (−0.142 + 0.989i)5-s + (−0.607 − 0.243i)6-s + (−0.0947 + 0.00904i)7-s + (−0.959 + 0.281i)8-s + (−0.237 + 0.520i)9-s + (−0.888 + 0.458i)10-s + (−0.154 − 0.635i)12-s + (−0.0623 − 0.0719i)14-s + (−0.271 − 0.595i)15-s + (−0.786 − 0.618i)16-s + (−0.561 + 0.108i)18-s + (−0.888 − 0.458i)20-s + (0.0489 − 0.0384i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.983 + 0.179i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.983 + 0.179i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9053764018\)
\(L(\frac12)\) \(\approx\) \(0.9053764018\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.580 - 0.814i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
good3 \( 1 + (0.550 - 0.353i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.0947 - 0.00904i)T + (0.981 - 0.189i)T^{2} \)
11 \( 1 + (-0.723 + 0.690i)T^{2} \)
13 \( 1 + (-0.0475 + 0.998i)T^{2} \)
17 \( 1 + (0.786 - 0.618i)T^{2} \)
19 \( 1 + (-0.981 - 0.189i)T^{2} \)
23 \( 1 + (0.0475 + 0.998i)T + (-0.995 + 0.0950i)T^{2} \)
29 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.0475 - 0.998i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.42 - 0.273i)T + (0.928 + 0.371i)T^{2} \)
43 \( 1 + (1.28 - 1.48i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.419 + 0.216i)T + (0.580 + 0.814i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.264 + 0.105i)T + (0.723 + 0.690i)T^{2} \)
71 \( 1 + (0.786 + 0.618i)T^{2} \)
73 \( 1 + (-0.723 - 0.690i)T^{2} \)
79 \( 1 + (0.888 - 0.458i)T^{2} \)
83 \( 1 + (-1.56 - 1.23i)T + (0.235 + 0.971i)T^{2} \)
89 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.37456711195852656842057057447, −9.417351046680702402067742338172, −8.309575196558193517379506093876, −7.73592404612025280017299134992, −6.65134610900054561348882991957, −6.29377254902315669083165962871, −5.20127657330402571181598939559, −4.54349474809989883555550674254, −3.42033073933053240026489432450, −2.53650563931009851583924348418, 0.67354295033210960361928083701, 1.89196042856425458901775615442, 3.31081499939402474702283839370, 4.21705857498273147830494054842, 5.14714197545847777195818047618, 5.84723359391626487005188204535, 6.60063456313598469780192450737, 7.82830560529974608738514269119, 8.824556701144667824162137996637, 9.488922928019484722554399904002

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