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Properties

Label 2-1340-1340.1119-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.289 + 0.957i$
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  + (0.723 − 0.690i)2-s + (−0.0135 + 0.0941i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.0552 + 0.0775i)6-s + (−0.370 − 1.52i)7-s + (−0.654 − 0.755i)8-s + (0.950 + 0.279i)9-s + (0.928 + 0.371i)10-s + (0.0934 + 0.0180i)12-s + (−1.32 − 0.849i)14-s + (−0.0913 + 0.0268i)15-s + (−0.995 − 0.0950i)16-s + (0.880 − 0.454i)18-s + (0.928 − 0.371i)20-s + (0.148 − 0.0142i)21-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (−0.0135 + 0.0941i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.0552 + 0.0775i)6-s + (−0.370 − 1.52i)7-s + (−0.654 − 0.755i)8-s + (0.950 + 0.279i)9-s + (0.928 + 0.371i)10-s + (0.0934 + 0.0180i)12-s + (−1.32 − 0.849i)14-s + (−0.0913 + 0.0268i)15-s + (−0.995 − 0.0950i)16-s + (0.880 − 0.454i)18-s + (0.928 − 0.371i)20-s + (0.148 − 0.0142i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657961737\)
\(L(\frac12)\) \(\approx\) \(1.657961737\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.723 + 0.690i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
good3 \( 1 + (0.0135 - 0.0941i)T + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (0.370 + 1.52i)T + (-0.888 + 0.458i)T^{2} \)
11 \( 1 + (0.327 + 0.945i)T^{2} \)
13 \( 1 + (0.786 - 0.618i)T^{2} \)
17 \( 1 + (0.995 - 0.0950i)T^{2} \)
19 \( 1 + (0.888 + 0.458i)T^{2} \)
23 \( 1 + (-0.786 - 0.618i)T + (0.235 + 0.971i)T^{2} \)
29 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.786 + 0.618i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.581 - 0.299i)T + (0.580 + 0.814i)T^{2} \)
43 \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-1.82 + 0.729i)T + (0.723 - 0.690i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.481 - 0.676i)T + (-0.327 + 0.945i)T^{2} \)
71 \( 1 + (0.995 + 0.0950i)T^{2} \)
73 \( 1 + (0.327 - 0.945i)T^{2} \)
79 \( 1 + (-0.928 - 0.371i)T^{2} \)
83 \( 1 + (0.469 + 0.0448i)T + (0.981 + 0.189i)T^{2} \)
89 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
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

−9.916319615219013084616751768071, −9.392537562658371503743152616306, −7.67216407105917189545895215761, −7.05870429624441269580714268563, −6.37646380291594576755533496223, −5.33673779380553255509678693859, −4.19357237249087932003956966625, −3.70590533560823925631323561304, −2.57040440714665386626720561538, −1.31644933836366099901219316223, 1.87288086627932863169180885589, 3.02792634149596701737575361724, 4.20998730881621060430491486919, 5.15351903540603469715416534891, 5.68127589111253566400131007263, 6.56332901284698754697408680081, 7.35099782271256559332432786420, 8.506176079900553408143923523593, 8.974343453973100331864704261946, 9.597834371922101568730773349492

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