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Properties

Label 2-1340-1340.659-c0-0-0
Degree $2$
Conductor $1340$
Sign $0.998 + 0.0490i$
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  + (−0.327 − 0.945i)2-s + (−0.653 − 1.43i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−1.13 + 1.08i)6-s + (−1.95 − 0.376i)7-s + (0.841 + 0.540i)8-s + (−0.963 + 1.11i)9-s + (0.580 + 0.814i)10-s + (1.39 + 0.720i)12-s + (0.283 + 1.97i)14-s + (1.02 + 1.18i)15-s + (0.235 − 0.971i)16-s + (1.36 + 0.546i)18-s + (0.580 − 0.814i)20-s + (0.737 + 3.04i)21-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (−0.653 − 1.43i)3-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−1.13 + 1.08i)6-s + (−1.95 − 0.376i)7-s + (0.841 + 0.540i)8-s + (−0.963 + 1.11i)9-s + (0.580 + 0.814i)10-s + (1.39 + 0.720i)12-s + (0.283 + 1.97i)14-s + (1.02 + 1.18i)15-s + (0.235 − 0.971i)16-s + (1.36 + 0.546i)18-s + (0.580 − 0.814i)20-s + (0.737 + 3.04i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.998 + 0.0490i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.998 + 0.0490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1377745596\)
\(L(\frac12)\) \(\approx\) \(0.1377745596\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 + 0.945i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
good3 \( 1 + (0.653 + 1.43i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (1.95 + 0.376i)T + (0.928 + 0.371i)T^{2} \)
11 \( 1 + (-0.0475 - 0.998i)T^{2} \)
13 \( 1 + (0.995 - 0.0950i)T^{2} \)
17 \( 1 + (-0.235 - 0.971i)T^{2} \)
19 \( 1 + (-0.928 + 0.371i)T^{2} \)
23 \( 1 + (-0.995 - 0.0950i)T + (0.981 + 0.189i)T^{2} \)
29 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.995 + 0.0950i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.0883 + 0.0353i)T + (0.723 - 0.690i)T^{2} \)
43 \( 1 + (0.264 - 1.83i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (1.03 - 1.44i)T + (-0.327 - 0.945i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (1.38 - 1.32i)T + (0.0475 - 0.998i)T^{2} \)
71 \( 1 + (-0.235 + 0.971i)T^{2} \)
73 \( 1 + (-0.0475 + 0.998i)T^{2} \)
79 \( 1 + (-0.580 - 0.814i)T^{2} \)
83 \( 1 + (-0.462 + 1.90i)T + (-0.888 - 0.458i)T^{2} \)
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

−9.870218796120561462538956603808, −9.148635914178771256733006237763, −8.002949254904411352968679868717, −7.38606005802106352678210454287, −6.70236039537464077944419039683, −5.94320079633574648342366671223, −4.45987914096870670961000381854, −3.37379193052877670450676378542, −2.67896435307456434736608912155, −1.03652046509493954394949484314, 0.17551713050757869359012690629, 3.34985294253609396317227837812, 3.85566648335290344004003635723, 4.97744813466369644834039999737, 5.53801953140480117283424215960, 6.57110066048506986023398814264, 7.13282040866237937191876764618, 8.452544443047057580657254683581, 9.166118042712494610820021244529, 9.593658782464571322590117522348

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