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Properties

Label 2-40e2-80.67-c1-0-8
Degree $2$
Conductor $1600$
Sign $0.922 - 0.386i$
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  + 0.207i·3-s + (−1.32 − 1.32i)7-s + 2.95·9-s + (2.39 + 2.39i)11-s − 4.20·13-s + (3.29 + 3.29i)17-s + (−0.838 − 0.838i)19-s + (0.274 − 0.274i)21-s + (2.67 − 2.67i)23-s + 1.23i·27-s + (2.55 − 2.55i)29-s + 6.23i·31-s + (−0.496 + 0.496i)33-s − 4.29·37-s − 0.871i·39-s + ⋯
L(s)  = 1  + 0.119i·3-s + (−0.499 − 0.499i)7-s + 0.985·9-s + (0.721 + 0.721i)11-s − 1.16·13-s + (0.798 + 0.798i)17-s + (−0.192 − 0.192i)19-s + (0.0598 − 0.0598i)21-s + (0.557 − 0.557i)23-s + 0.237i·27-s + (0.474 − 0.474i)29-s + 1.12i·31-s + (−0.0864 + 0.0864i)33-s − 0.706·37-s − 0.139i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744316529\)
\(L(\frac12)\) \(\approx\) \(1.744316529\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.207iT - 3T^{2} \)
7 \( 1 + (1.32 + 1.32i)T + 7iT^{2} \)
11 \( 1 + (-2.39 - 2.39i)T + 11iT^{2} \)
13 \( 1 + 4.20T + 13T^{2} \)
17 \( 1 + (-3.29 - 3.29i)T + 17iT^{2} \)
19 \( 1 + (0.838 + 0.838i)T + 19iT^{2} \)
23 \( 1 + (-2.67 + 2.67i)T - 23iT^{2} \)
29 \( 1 + (-2.55 + 2.55i)T - 29iT^{2} \)
31 \( 1 - 6.23iT - 31T^{2} \)
37 \( 1 + 4.29T + 37T^{2} \)
41 \( 1 - 7.06iT - 41T^{2} \)
43 \( 1 - 9.43T + 43T^{2} \)
47 \( 1 + (-8.31 + 8.31i)T - 47iT^{2} \)
53 \( 1 - 7.66iT - 53T^{2} \)
59 \( 1 + (-7.07 + 7.07i)T - 59iT^{2} \)
61 \( 1 + (-8.74 - 8.74i)T + 61iT^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (4.79 + 4.79i)T + 73iT^{2} \)
79 \( 1 - 8.69T + 79T^{2} \)
83 \( 1 - 4.14iT - 83T^{2} \)
89 \( 1 + 0.548T + 89T^{2} \)
97 \( 1 + (-12.2 - 12.2i)T + 97iT^{2} \)
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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

−9.598653929265966754054961348530, −8.796558012601540346264547832104, −7.67032330999822943806412350070, −7.03793142460717353047131944382, −6.47096128588435489276499510871, −5.19661380018714193455346692315, −4.37363704745455621133377309165, −3.64942659790620596289326809066, −2.35660439429643076215987952364, −1.07823303688914404874885273870, 0.864270740685157191397693397052, 2.28407046801419592264631645969, 3.31399365734943909742154085130, 4.29493580165783675186553078527, 5.32040693233994584330583734048, 6.09585099010567880537645384391, 7.11862023486546184750004863451, 7.52476866685623129433390356496, 8.711200516884753290819539861409, 9.447725203438155570988461290730

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