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Properties

Label 2-1340-1340.199-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.998 + 0.0536i$
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  + (0.0475 − 0.998i)2-s + (1.91 + 0.560i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.651 − 1.88i)6-s + (−0.419 + 0.216i)7-s + (−0.142 + 0.989i)8-s + (2.49 + 1.60i)9-s + (0.723 + 0.690i)10-s + (−1.84 − 0.739i)12-s + (0.195 + 0.428i)14-s + (−1.67 + 1.07i)15-s + (0.981 + 0.189i)16-s + (1.71 − 2.41i)18-s + (0.723 − 0.690i)20-s + (−0.921 + 0.177i)21-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (1.91 + 0.560i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.651 − 1.88i)6-s + (−0.419 + 0.216i)7-s + (−0.142 + 0.989i)8-s + (2.49 + 1.60i)9-s + (0.723 + 0.690i)10-s + (−1.84 − 0.739i)12-s + (0.195 + 0.428i)14-s + (−1.67 + 1.07i)15-s + (0.981 + 0.189i)16-s + (1.71 − 2.41i)18-s + (0.723 − 0.690i)20-s + (−0.921 + 0.177i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.998 + 0.0536i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.998 + 0.0536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.633846788\)
\(L(\frac12)\) \(\approx\) \(1.633846788\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 + 0.998i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
good3 \( 1 + (-1.91 - 0.560i)T + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.419 - 0.216i)T + (0.580 - 0.814i)T^{2} \)
11 \( 1 + (0.786 - 0.618i)T^{2} \)
13 \( 1 + (-0.235 + 0.971i)T^{2} \)
17 \( 1 + (-0.981 + 0.189i)T^{2} \)
19 \( 1 + (-0.580 - 0.814i)T^{2} \)
23 \( 1 + (0.235 + 0.971i)T + (-0.888 + 0.458i)T^{2} \)
29 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.235 - 0.971i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.911 + 1.28i)T + (-0.327 + 0.945i)T^{2} \)
43 \( 1 + (-0.481 + 1.05i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-1.34 + 1.28i)T + (0.0475 - 0.998i)T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.428 + 1.23i)T + (-0.786 - 0.618i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (0.786 + 0.618i)T^{2} \)
79 \( 1 + (-0.723 - 0.690i)T^{2} \)
83 \( 1 + (1.74 + 0.336i)T + (0.928 + 0.371i)T^{2} \)
89 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
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

−9.864345993775856350316747401556, −8.948127351062154216107805453471, −8.567357761567573740249737646446, −7.67755049157559416880092866877, −6.87530996867433118504014683890, −5.17347594222258903793176222830, −4.05523753750937459663775156529, −3.58970535359850048843922717869, −2.78143762045118689275427371786, −2.02031653844003743367190210235, 1.30865228374160533406635966033, 2.94081092295622857594144116113, 3.88143032262980671172947850495, 4.41698691631260062343784714789, 5.85735501391946376857241456860, 6.96681049362851127996108767861, 7.54916473280453195311419106771, 8.143726174330452804189952725543, 8.708358197582639679413876444457, 9.553268491754624562243167852013

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