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Properties

Label 2-1340-1340.759-c0-0-0
Degree $2$
Conductor $1340$
Sign $0.874 - 0.484i$
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  + (0.959 − 0.281i)2-s + (0.239 + 1.66i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.698 + 1.53i)6-s + (−0.273 + 0.0801i)7-s + (0.654 − 0.755i)8-s + (−1.75 + 0.515i)9-s + (0.142 − 0.989i)10-s + (1.10 + 1.27i)12-s + (−0.239 + 0.153i)14-s + (1.61 + 0.474i)15-s + (0.415 − 0.909i)16-s + (−1.54 + 0.989i)18-s + (−0.142 − 0.989i)20-s + (−0.198 − 0.435i)21-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.239 + 1.66i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.698 + 1.53i)6-s + (−0.273 + 0.0801i)7-s + (0.654 − 0.755i)8-s + (−1.75 + 0.515i)9-s + (0.142 − 0.989i)10-s + (1.10 + 1.27i)12-s + (−0.239 + 0.153i)14-s + (1.61 + 0.474i)15-s + (0.415 − 0.909i)16-s + (−1.54 + 0.989i)18-s + (−0.142 − 0.989i)20-s + (−0.198 − 0.435i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.098284599\)
\(L(\frac12)\) \(\approx\) \(2.098284599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
23 \( 1 + (-0.284 - 1.97i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + 0.284T + T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
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

−10.02251399129094006728397294708, −9.285390227269660257079635790560, −8.629732265957840351955935745365, −7.38213286882595266468334125389, −6.08544397633681139227299991384, −5.27177873962106412347951079396, −4.88511075042708211955391201321, −3.81768790722258867068861925561, −3.27726434122538814357413310050, −1.85904259067954817075697877778, 1.71912996604964882187123301627, 2.64176817407235728848404176926, 3.32473994108461950059952230535, 4.79784510838149282760525064306, 5.99591196274761968226706605654, 6.58489653745690333753275263586, 6.92285369432911246553815935898, 7.88203869497545833431434471942, 8.461451816138033903573538827138, 9.820054516841470804231770109083

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