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Properties

Label 2-40e2-80.27-c1-0-11
Degree $2$
Conductor $1600$
Sign $0.604 - 0.796i$
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  − 3.25·3-s + (2.54 + 2.54i)7-s + 7.61·9-s + (0.462 − 0.462i)11-s + 1.33i·13-s + (2.37 + 2.37i)17-s + (2.69 − 2.69i)19-s + (−8.27 − 8.27i)21-s + (−2.10 + 2.10i)23-s − 15.0·27-s + (1.97 + 1.97i)29-s − 7.03i·31-s + (−1.50 + 1.50i)33-s − 7.81i·37-s − 4.34i·39-s + ⋯
L(s)  = 1  − 1.88·3-s + (0.960 + 0.960i)7-s + 2.53·9-s + (0.139 − 0.139i)11-s + 0.370i·13-s + (0.575 + 0.575i)17-s + (0.618 − 0.618i)19-s + (−1.80 − 1.80i)21-s + (−0.438 + 0.438i)23-s − 2.89·27-s + (0.367 + 0.367i)29-s − 1.26i·31-s + (−0.262 + 0.262i)33-s − 1.28i·37-s − 0.696i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.003961620\)
\(L(\frac12)\) \(\approx\) \(1.003961620\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.25T + 3T^{2} \)
7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 + (-0.462 + 0.462i)T - 11iT^{2} \)
13 \( 1 - 1.33iT - 13T^{2} \)
17 \( 1 + (-2.37 - 2.37i)T + 17iT^{2} \)
19 \( 1 + (-2.69 + 2.69i)T - 19iT^{2} \)
23 \( 1 + (2.10 - 2.10i)T - 23iT^{2} \)
29 \( 1 + (-1.97 - 1.97i)T + 29iT^{2} \)
31 \( 1 + 7.03iT - 31T^{2} \)
37 \( 1 + 7.81iT - 37T^{2} \)
41 \( 1 - 2.17iT - 41T^{2} \)
43 \( 1 + 3.10iT - 43T^{2} \)
47 \( 1 + (-0.0727 + 0.0727i)T - 47iT^{2} \)
53 \( 1 - 0.719T + 53T^{2} \)
59 \( 1 + (-8.67 - 8.67i)T + 59iT^{2} \)
61 \( 1 + (7.10 - 7.10i)T - 61iT^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (0.905 + 0.905i)T + 73iT^{2} \)
79 \( 1 + 3.90T + 79T^{2} \)
83 \( 1 - 6.02T + 83T^{2} \)
89 \( 1 + 7.46T + 89T^{2} \)
97 \( 1 + (-3.74 - 3.74i)T + 97iT^{2} \)
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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

−9.698278705318771562725641253840, −8.798053454544392954869369315400, −7.73775974227034847744112780906, −6.99648946339114881524908025761, −5.93753003230493243964073066423, −5.59073259529243583108737441036, −4.81285413444097111362612858091, −3.92994067594803424756672499245, −2.13528872047573130113659107369, −1.00093520994289727455121748377, 0.68753709649051758594897500617, 1.57727761619813214359486376043, 3.57050048510989273594187907560, 4.69980469134094898567875202828, 5.02596684953034616374062418252, 5.99971960784572010112452963813, 6.79080769212892132408962738910, 7.50983843552515434090850289118, 8.232987820852809302968524056575, 9.789082945564528083460268741155

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