Properties

Label 2-1340-1340.859-c0-0-1
Degree $2$
Conductor $1340$
Sign $-0.154 + 0.987i$
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  + (−0.928 − 0.371i)2-s + (0.947 − 1.09i)3-s + (0.723 + 0.690i)4-s + (0.841 − 0.540i)5-s + (−1.28 + 0.663i)6-s + (−0.514 + 0.404i)7-s + (−0.415 − 0.909i)8-s + (−0.155 − 1.08i)9-s + (−0.981 + 0.189i)10-s + (1.44 − 0.137i)12-s + (0.627 − 0.184i)14-s + (0.205 − 1.43i)15-s + (0.0475 + 0.998i)16-s + (−0.258 + 1.06i)18-s + (0.981 + 0.189i)20-s + (−0.0450 + 0.945i)21-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)2-s + (0.947 − 1.09i)3-s + (0.723 + 0.690i)4-s + (0.841 − 0.540i)5-s + (−1.28 + 0.663i)6-s + (−0.514 + 0.404i)7-s + (−0.415 − 0.909i)8-s + (−0.155 − 1.08i)9-s + (−0.981 + 0.189i)10-s + (1.44 − 0.137i)12-s + (0.627 − 0.184i)14-s + (0.205 − 1.43i)15-s + (0.0475 + 0.998i)16-s + (−0.258 + 1.06i)18-s + (0.981 + 0.189i)20-s + (−0.0450 + 0.945i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.154 + 0.987i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.154 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056101330\)
\(L(\frac12)\) \(\approx\) \(1.056101330\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.928 + 0.371i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
good3 \( 1 + (-0.947 + 1.09i)T + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.514 - 0.404i)T + (0.235 - 0.971i)T^{2} \)
11 \( 1 + (-0.580 - 0.814i)T^{2} \)
13 \( 1 + (0.327 - 0.945i)T^{2} \)
17 \( 1 + (-0.0475 + 0.998i)T^{2} \)
19 \( 1 + (-0.235 - 0.971i)T^{2} \)
23 \( 1 + (0.327 + 0.945i)T + (-0.786 + 0.618i)T^{2} \)
29 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.327 + 0.945i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.273 - 1.12i)T + (-0.888 + 0.458i)T^{2} \)
43 \( 1 + (-0.452 - 0.132i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (-1.95 - 0.376i)T + (0.928 + 0.371i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (1.49 - 0.770i)T + (0.580 - 0.814i)T^{2} \)
71 \( 1 + (-0.0475 - 0.998i)T^{2} \)
73 \( 1 + (-0.580 + 0.814i)T^{2} \)
79 \( 1 + (-0.981 + 0.189i)T^{2} \)
83 \( 1 + (-0.0748 - 1.57i)T + (-0.995 + 0.0950i)T^{2} \)
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

−9.357630255618278097790767441107, −8.868007028274146832211983116418, −8.087951001731615730398391744332, −7.45950065590377298599665753795, −6.45023234328095257293406434714, −5.87594878176282160926876090083, −4.17423726255157274003290411320, −2.73642430919656068769581916318, −2.30482106585476922562708740002, −1.16276497683607374474389941344, 1.83843926789901982320900795172, 2.95180229866564744406562298617, 3.74968589787967634191178070135, 5.16216366605543521266200387338, 5.98056529175461133937868609308, 7.01117518266900097902647913982, 7.62854935930050477916549023373, 8.838527773886037484090754619184, 9.221503414530376650545615460087, 9.823695318583474233487975340441

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