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Properties

Label 2-1340-1340.359-c0-0-0
Degree $2$
Conductor $1340$
Sign $-0.366 - 0.930i$
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  + (−0.415 + 0.909i)2-s + (1.10 − 0.708i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (0.186 + 1.29i)6-s + (−0.698 + 1.53i)7-s + (0.959 − 0.281i)8-s + (0.297 − 0.650i)9-s + (−0.841 − 0.540i)10-s + (−1.25 − 0.368i)12-s + (−1.10 − 1.27i)14-s + (0.544 + 1.19i)15-s + (−0.142 + 0.989i)16-s + (0.468 + 0.540i)18-s + (0.841 − 0.540i)20-s + (0.313 + 2.18i)21-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (1.10 − 0.708i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (0.186 + 1.29i)6-s + (−0.698 + 1.53i)7-s + (0.959 − 0.281i)8-s + (0.297 − 0.650i)9-s + (−0.841 − 0.540i)10-s + (−1.25 − 0.368i)12-s + (−1.10 − 1.27i)14-s + (0.544 + 1.19i)15-s + (−0.142 + 0.989i)16-s + (0.468 + 0.540i)18-s + (0.841 − 0.540i)20-s + (0.313 + 2.18i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.366 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9961838149\)
\(L(\frac12)\) \(\approx\) \(0.9961838149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
good3 \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
23 \( 1 + (1.68 - 1.08i)T + (0.415 - 0.909i)T^{2} \)
29 \( 1 - 1.68T + T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.738410079525235117771735994661, −9.019670361654395100707754341353, −8.345691859803245949595039562093, −7.70661919881539050429568047214, −6.91275668637051797485612141722, −6.16389050968073596787363062038, −5.49350496762291365410340434107, −3.88267779245881458609974455984, −2.79334091630481135564566988509, −2.01047483585881177405901166360, 0.868054638299173842371867989291, 2.43492863134600800633385709667, 3.51404890565090870566505812981, 4.17574342543011710334705388280, 4.65298040746243484013953457752, 6.37213576698966090165840412931, 7.66353616013503381656547498414, 8.180286786410928059597640654978, 8.913651553011596806111418280971, 9.683236585880899134082892783024

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