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Properties

Label 2-1340-1340.1179-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.215 + 0.976i$
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  + (−0.654 − 0.755i)2-s + (−0.118 + 0.258i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (0.273 − 0.0801i)6-s + (−0.544 − 0.627i)7-s + (0.841 − 0.540i)8-s + (0.601 + 0.694i)9-s + (0.415 + 0.909i)10-s + (−0.239 − 0.153i)12-s + (−0.118 + 0.822i)14-s + (0.186 − 0.215i)15-s + (−0.959 − 0.281i)16-s + (0.130 − 0.909i)18-s + (0.415 − 0.909i)20-s + (0.226 − 0.0666i)21-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.118 + 0.258i)3-s + (−0.142 + 0.989i)4-s + (−0.959 − 0.281i)5-s + (0.273 − 0.0801i)6-s + (−0.544 − 0.627i)7-s + (0.841 − 0.540i)8-s + (0.601 + 0.694i)9-s + (0.415 + 0.909i)10-s + (−0.239 − 0.153i)12-s + (−0.118 + 0.822i)14-s + (0.186 − 0.215i)15-s + (−0.959 − 0.281i)16-s + (0.130 − 0.909i)18-s + (0.415 − 0.909i)20-s + (0.226 − 0.0666i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.215 + 0.976i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5727935250\)
\(L(\frac12)\) \(\approx\) \(0.5727935250\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
23 \( 1 + (-0.830 + 1.81i)T + (-0.654 - 0.755i)T^{2} \)
29 \( 1 - 0.830T + T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
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

−9.852892408366569427509707548164, −8.750315783571721425668496875966, −8.291356660113768658764287612551, −7.22435720204754265333316438798, −6.85204321415966673403878429518, −5.05819346037603069628150040692, −4.26142004546565624730702660838, −3.58181204196126934588819875296, −2.37693789755653887489209194237, −0.74451456904892261038976933828, 1.19287265829250625389973355446, 2.91617901573824883390039209496, 4.05495391479181019752332973112, 5.15017912824784781835885062752, 6.17278569223833392591624626520, 6.81196904703588942082431262964, 7.52045619483204121394169897812, 8.231052391841609841293744781082, 9.236967183549815732429829380806, 9.636428733347522929041511081309

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