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Properties

Label 2-40e2-80.43-c1-0-31
Degree $2$
Conductor $1600$
Sign $-0.934 + 0.356i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70i·3-s + (−1.60 + 1.60i)7-s − 4.33·9-s + (3.90 − 3.90i)11-s + 1.48·13-s + (4.26 − 4.26i)17-s + (0.162 − 0.162i)19-s + (4.35 + 4.35i)21-s + (−5.87 − 5.87i)23-s + 3.60i·27-s + (1.48 + 1.48i)29-s + 6.78i·31-s + (−10.5 − 10.5i)33-s − 6.13·37-s − 4.02i·39-s + ⋯
L(s)  = 1  − 1.56i·3-s + (−0.608 + 0.608i)7-s − 1.44·9-s + (1.17 − 1.17i)11-s + 0.412·13-s + (1.03 − 1.03i)17-s + (0.0373 − 0.0373i)19-s + (0.950 + 0.950i)21-s + (−1.22 − 1.22i)23-s + 0.694i·27-s + (0.276 + 0.276i)29-s + 1.21i·31-s + (−1.84 − 1.84i)33-s − 1.00·37-s − 0.644i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.934 + 0.356i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.385665005\)
\(L(\frac12)\) \(\approx\) \(1.385665005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 2.70iT - 3T^{2} \)
7 \( 1 + (1.60 - 1.60i)T - 7iT^{2} \)
11 \( 1 + (-3.90 + 3.90i)T - 11iT^{2} \)
13 \( 1 - 1.48T + 13T^{2} \)
17 \( 1 + (-4.26 + 4.26i)T - 17iT^{2} \)
19 \( 1 + (-0.162 + 0.162i)T - 19iT^{2} \)
23 \( 1 + (5.87 + 5.87i)T + 23iT^{2} \)
29 \( 1 + (-1.48 - 1.48i)T + 29iT^{2} \)
31 \( 1 - 6.78iT - 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 2.75iT - 41T^{2} \)
43 \( 1 - 3.39T + 43T^{2} \)
47 \( 1 + (9.44 + 9.44i)T + 47iT^{2} \)
53 \( 1 + 1.54iT - 53T^{2} \)
59 \( 1 + (2.53 + 2.53i)T + 59iT^{2} \)
61 \( 1 + (-0.600 + 0.600i)T - 61iT^{2} \)
67 \( 1 + 8.14T + 67T^{2} \)
71 \( 1 - 4.55T + 71T^{2} \)
73 \( 1 + (-4.84 + 4.84i)T - 73iT^{2} \)
79 \( 1 - 0.455T + 79T^{2} \)
83 \( 1 - 2.84iT - 83T^{2} \)
89 \( 1 + 1.91T + 89T^{2} \)
97 \( 1 + (1.73 - 1.73i)T - 97iT^{2} \)
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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

−8.728471483245789525010797266284, −8.369938136094331871914558606965, −7.34100918590045976717743819491, −6.50942257138015494487867393549, −6.17814540693256460774773597755, −5.22642094369514100904494868979, −3.62431815774421450628765076866, −2.83901270002440539710120465117, −1.63536112308338226411831334641, −0.55669183016719040713138537849, 1.60465801435892763602165412452, 3.33161427496629447239037472530, 3.93230074291751513868898490297, 4.45466014076562661761401783788, 5.64386337682242873698932242089, 6.34959718749901503739030595727, 7.42353832297949596741875656556, 8.301520491136726232890197154713, 9.413673795364495450202389269911, 9.743000906987695598168838547109

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