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Properties

Label 2-1340-1340.819-c0-0-0
Degree $2$
Conductor $1340$
Sign $0.776 + 0.629i$
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  + (−0.841 + 0.540i)2-s + (0.797 − 0.234i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (−0.544 + 0.627i)6-s + (1.61 − 1.03i)7-s + (0.142 + 0.989i)8-s + (−0.260 + 0.167i)9-s + (0.959 + 0.281i)10-s + (0.118 − 0.822i)12-s + (−0.797 + 1.74i)14-s + (−0.698 − 0.449i)15-s + (−0.654 − 0.755i)16-s + (0.128 − 0.281i)18-s + (−0.959 + 0.281i)20-s + (1.04 − 1.20i)21-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.797 − 0.234i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (−0.544 + 0.627i)6-s + (1.61 − 1.03i)7-s + (0.142 + 0.989i)8-s + (−0.260 + 0.167i)9-s + (0.959 + 0.281i)10-s + (0.118 − 0.822i)12-s + (−0.797 + 1.74i)14-s + (−0.698 − 0.449i)15-s + (−0.654 − 0.755i)16-s + (0.128 − 0.281i)18-s + (−0.959 + 0.281i)20-s + (1.04 − 1.20i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.776 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9627464456\)
\(L(\frac12)\) \(\approx\) \(0.9627464456\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
good3 \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
23 \( 1 + (-1.91 + 0.563i)T + (0.841 - 0.540i)T^{2} \)
29 \( 1 + 1.91T + T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
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

−9.297662807818671818315037939369, −8.704646386905465324008553197675, −8.124589169932556092102119830245, −7.53048405716189516805179848868, −7.01608231555601468599055224248, −5.36234738418380974717863173767, −4.86688688291765846610342329801, −3.70218437840882148354463595809, −2.10420220310407372596229383520, −1.08777665597053889954361323206, 1.74241629228153263470268930610, 2.72800131668859013360504091654, 3.44081121074261379850087192991, 4.56192894899479644408461115458, 5.76325511425243920972939249207, 7.11562685679624382780453747160, 7.76119293454567142757828986136, 8.439158373451851473666177744927, 8.968495122196960908961342034296, 9.687339251963944857434514737834

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