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Properties

Label 2-1340-1340.639-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.463 + 0.886i$
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  + (−0.786 − 0.618i)2-s + (−0.308 − 0.356i)3-s + (0.235 + 0.971i)4-s + (0.841 + 0.540i)5-s + (0.0224 + 0.470i)6-s + (1.82 − 0.729i)7-s + (0.415 − 0.909i)8-s + (0.110 − 0.769i)9-s + (−0.327 − 0.945i)10-s + (0.273 − 0.384i)12-s + (−1.88 − 0.553i)14-s + (−0.0671 − 0.466i)15-s + (−0.888 + 0.458i)16-s + (−0.562 + 0.536i)18-s + (−0.327 + 0.945i)20-s + (−0.823 − 0.424i)21-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (−0.308 − 0.356i)3-s + (0.235 + 0.971i)4-s + (0.841 + 0.540i)5-s + (0.0224 + 0.470i)6-s + (1.82 − 0.729i)7-s + (0.415 − 0.909i)8-s + (0.110 − 0.769i)9-s + (−0.327 − 0.945i)10-s + (0.273 − 0.384i)12-s + (−1.88 − 0.553i)14-s + (−0.0671 − 0.466i)15-s + (−0.888 + 0.458i)16-s + (−0.562 + 0.536i)18-s + (−0.327 + 0.945i)20-s + (−0.823 − 0.424i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.463 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9370281833\)
\(L(\frac12)\) \(\approx\) \(0.9370281833\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 + 0.618i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
good3 \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (-1.82 + 0.729i)T + (0.723 - 0.690i)T^{2} \)
11 \( 1 + (0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.981 - 0.189i)T^{2} \)
17 \( 1 + (0.888 + 0.458i)T^{2} \)
19 \( 1 + (-0.723 - 0.690i)T^{2} \)
23 \( 1 + (0.981 - 0.189i)T + (0.928 - 0.371i)T^{2} \)
29 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.981 + 0.189i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.44 + 1.37i)T + (0.0475 + 0.998i)T^{2} \)
43 \( 1 + (1.38 - 0.407i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.379 - 1.09i)T + (-0.786 - 0.618i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.0800 - 1.68i)T + (-0.995 + 0.0950i)T^{2} \)
71 \( 1 + (0.888 - 0.458i)T^{2} \)
73 \( 1 + (0.995 - 0.0950i)T^{2} \)
79 \( 1 + (0.327 + 0.945i)T^{2} \)
83 \( 1 + (1.65 - 0.850i)T + (0.580 - 0.814i)T^{2} \)
89 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
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

−9.903394001878290174512260501852, −8.863827310896890899651535177656, −8.142478998972175886849229986166, −7.27382533682191560119708994816, −6.70775366134108067597990334141, −5.55394969138543059590933801116, −4.43371505778285903507698154880, −3.38239197888715238891784540289, −1.97931842692408428399381407032, −1.30335897707213693143106926854, 1.61959293069062669040245114525, 2.19853831621678677445554647746, 4.57836253035020715157599519817, 5.08686924464944994445280046867, 5.66262375961190768546213027174, 6.62128765523855159394623128824, 7.980102491943747468539174083159, 8.209807374918695189722728923831, 8.988999676267423509636355927164, 10.02308110009126832175525740630

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