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Properties

Label 2-1340-1340.59-c0-0-1
Degree $2$
Conductor $1340$
Sign $-0.770 - 0.637i$
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  + (−0.142 + 0.989i)2-s + (1.25 + 1.45i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−1.61 + 1.03i)6-s + (0.186 − 1.29i)7-s + (0.415 − 0.909i)8-s + (−0.381 + 2.65i)9-s + (−0.654 + 0.755i)10-s + (−0.797 − 1.74i)12-s + (1.25 + 0.368i)14-s + (0.273 + 1.89i)15-s + (0.841 + 0.540i)16-s + (−2.57 − 0.755i)18-s + (−0.654 − 0.755i)20-s + (2.11 − 1.35i)21-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (1.25 + 1.45i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−1.61 + 1.03i)6-s + (0.186 − 1.29i)7-s + (0.415 − 0.909i)8-s + (−0.381 + 2.65i)9-s + (−0.654 + 0.755i)10-s + (−0.797 − 1.74i)12-s + (1.25 + 0.368i)14-s + (0.273 + 1.89i)15-s + (0.841 + 0.540i)16-s + (−2.57 − 0.755i)18-s + (−0.654 − 0.755i)20-s + (2.11 − 1.35i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540764857\)
\(L(\frac12)\) \(\approx\) \(1.540764857\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
23 \( 1 + (1.30 + 1.51i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + 1.30T + T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
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

−10.04890251650440941113510932275, −9.315580528653929534030609038429, −8.559698078331888492433481949648, −7.79074846728011682042402020265, −7.07410992621583801906019399682, −5.96134541112029831664264314457, −4.95568827289558927338106796437, −4.15088786879053605113702831136, −3.51698363055166889206363742775, −2.13642311489121806373498448498, 1.46200929495042277107614036154, 2.08470464792463469442984542044, 2.81153546226415597114535073578, 3.91845571211874175957637707497, 5.47658529234958366494354109599, 6.04511517604337619474463104376, 7.42825811801257336430660679079, 8.137123910912561214351853807072, 8.830003279601612608997730801283, 9.308603119438470509443561941155

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