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Properties

Label 2-1340-1340.959-c0-0-1
Degree $2$
Conductor $1340$
Sign $-0.367 + 0.929i$
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  + (0.981 − 0.189i)2-s + (0.771 − 1.68i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.437 − 1.80i)6-s + (−0.379 + 1.09i)7-s + (0.841 − 0.540i)8-s + (−1.60 − 1.84i)9-s + (−0.995 − 0.0950i)10-s + (0.0883 − 1.85i)12-s + (−0.165 + 1.14i)14-s + (−1.21 + 1.40i)15-s + (0.723 − 0.690i)16-s + (−1.92 − 1.51i)18-s + (−0.995 + 0.0950i)20-s + (1.55 + 1.48i)21-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (0.771 − 1.68i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.437 − 1.80i)6-s + (−0.379 + 1.09i)7-s + (0.841 − 0.540i)8-s + (−1.60 − 1.84i)9-s + (−0.995 − 0.0950i)10-s + (0.0883 − 1.85i)12-s + (−0.165 + 1.14i)14-s + (−1.21 + 1.40i)15-s + (0.723 − 0.690i)16-s + (−1.92 − 1.51i)18-s + (−0.995 + 0.0950i)20-s + (1.55 + 1.48i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.367 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.035312276\)
\(L(\frac12)\) \(\approx\) \(2.035312276\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
good3 \( 1 + (-0.771 + 1.68i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.379 - 1.09i)T + (-0.786 - 0.618i)T^{2} \)
11 \( 1 + (0.888 - 0.458i)T^{2} \)
13 \( 1 + (-0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.723 - 0.690i)T^{2} \)
19 \( 1 + (0.786 - 0.618i)T^{2} \)
23 \( 1 + (0.580 + 0.814i)T + (-0.327 + 0.945i)T^{2} \)
29 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.580 - 0.814i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.39 + 1.09i)T + (0.235 - 0.971i)T^{2} \)
43 \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.0947 - 0.00904i)T + (0.981 - 0.189i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.452 - 1.86i)T + (-0.888 - 0.458i)T^{2} \)
71 \( 1 + (-0.723 + 0.690i)T^{2} \)
73 \( 1 + (0.888 + 0.458i)T^{2} \)
79 \( 1 + (0.995 + 0.0950i)T^{2} \)
83 \( 1 + (0.473 - 0.451i)T + (0.0475 - 0.998i)T^{2} \)
89 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
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

−9.212189991208225350602653694623, −8.559485005086962297449269237323, −7.79150618113953051908941552545, −7.05493202397324897024254347689, −6.34724151345007728766777632539, −5.54724071130078037366608276028, −4.27570466286039538808049918501, −3.07520645074558744292220724623, −2.57917622167584488799294165157, −1.30936463340297473600314937289, 2.61935562378437790235145208441, 3.45935067600010548472763853360, 4.11977161890727877762688481147, 4.48621209059365182027184670735, 5.63859633325427728129909804824, 6.78265300872259242375560163896, 7.80176266494574364785992100338, 8.180083645145535716010838187244, 9.441300287614448224549823657966, 10.19421048327235030749006575421

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