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Properties

Label 2-1340-1340.419-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.233 + 0.972i$
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  + (−0.995 + 0.0950i)2-s + (1.65 − 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (−1.03 − 1.44i)7-s + (−0.959 + 0.281i)8-s + (1.18 − 2.59i)9-s + (0.0475 − 0.998i)10-s + (1.42 − 1.35i)12-s + (1.16 + 1.34i)14-s + (0.815 + 1.78i)15-s + (0.928 − 0.371i)16-s + (−0.934 + 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (1.65 − 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (−1.03 − 1.44i)7-s + (−0.959 + 0.281i)8-s + (1.18 − 2.59i)9-s + (0.0475 − 0.998i)10-s + (1.42 − 1.35i)12-s + (1.16 + 1.34i)14-s + (0.815 + 1.78i)15-s + (0.928 − 0.371i)16-s + (−0.934 + 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.233 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073864390\)
\(L(\frac12)\) \(\approx\) \(1.073864390\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 - 0.0950i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
good3 \( 1 + (-1.65 + 1.06i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (1.03 + 1.44i)T + (-0.327 + 0.945i)T^{2} \)
11 \( 1 + (-0.235 - 0.971i)T^{2} \)
13 \( 1 + (0.888 - 0.458i)T^{2} \)
17 \( 1 + (-0.928 - 0.371i)T^{2} \)
19 \( 1 + (0.327 + 0.945i)T^{2} \)
23 \( 1 + (-0.888 - 0.458i)T + (0.580 + 0.814i)T^{2} \)
29 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.154 + 0.445i)T + (-0.786 + 0.618i)T^{2} \)
43 \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.223 + 0.175i)T + (0.235 - 0.971i)T^{2} \)
71 \( 1 + (-0.928 + 0.371i)T^{2} \)
73 \( 1 + (-0.235 + 0.971i)T^{2} \)
79 \( 1 + (-0.0475 + 0.998i)T^{2} \)
83 \( 1 + (-1.07 + 0.431i)T + (0.723 - 0.690i)T^{2} \)
89 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
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

−9.496763479926060930731230314038, −8.852573793700857162965965889661, −7.68820454954955578779628545997, −7.49264211710613490105836859356, −6.76796082059093656189184068485, −6.28479752346199474101115838870, −3.81587416120543448595321320708, −3.22313223592870771826412779560, −2.40561407232622758812268283252, −1.07714906912893383293377559048, 1.96584864339041134867457501641, 2.82741809109711057372450659073, 3.56712856721854168055968432628, 4.79869818095111307125041809802, 5.79753031714344636289117987391, 7.08574858737858474064847364633, 8.159691733931486176084628408747, 8.599125333167721882014994460527, 9.181968234506796259392715971823, 9.572366840995407021344081289525

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