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Properties

Label 2-40e2-16.13-c1-0-32
Degree $2$
Conductor $1600$
Sign $-0.635 - 0.772i$
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  + (−1.86 − 1.86i)3-s − 3.61i·7-s + 3.92i·9-s + (0.0947 − 0.0947i)11-s + (−2.59 − 2.59i)13-s + 1.89·17-s + (−2.16 − 2.16i)19-s + (−6.72 + 6.72i)21-s − 5.08i·23-s + (1.71 − 1.71i)27-s + (−1.25 − 1.25i)29-s + 1.27·31-s − 0.352·33-s + (−2.25 + 2.25i)37-s + 9.65i·39-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)3-s − 1.36i·7-s + 1.30i·9-s + (0.0285 − 0.0285i)11-s + (−0.719 − 0.719i)13-s + 0.460·17-s + (−0.496 − 0.496i)19-s + (−1.46 + 1.46i)21-s − 1.05i·23-s + (0.329 − 0.329i)27-s + (−0.233 − 0.233i)29-s + 0.228·31-s − 0.0613·33-s + (−0.370 + 0.370i)37-s + 1.54i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.635 - 0.772i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.635 - 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4997163641\)
\(L(\frac12)\) \(\approx\) \(0.4997163641\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.86i)T + 3iT^{2} \)
7 \( 1 + 3.61iT - 7T^{2} \)
11 \( 1 + (-0.0947 + 0.0947i)T - 11iT^{2} \)
13 \( 1 + (2.59 + 2.59i)T + 13iT^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 + (2.16 + 2.16i)T + 19iT^{2} \)
23 \( 1 + 5.08iT - 23T^{2} \)
29 \( 1 + (1.25 + 1.25i)T + 29iT^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + (2.25 - 2.25i)T - 37iT^{2} \)
41 \( 1 + 8.52iT - 41T^{2} \)
43 \( 1 + (1.61 - 1.61i)T - 43iT^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 + (-5.67 + 5.67i)T - 53iT^{2} \)
59 \( 1 + (7.81 - 7.81i)T - 59iT^{2} \)
61 \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \)
67 \( 1 + (6.29 + 6.29i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 + (3.75 + 3.75i)T + 83iT^{2} \)
89 \( 1 + 3.98iT - 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−8.730589173249596107059957682509, −7.75451197543324749475213424426, −7.19180389832417585059005055795, −6.63785126145814311093721532863, −5.72480376735023263251113616734, −4.88768324074217184466301306509, −3.90890113543241355481127003605, −2.50725777053962396114059117221, −1.13100064883006902634599966392, −0.25009153185942262153584084543, 1.88182067638836927125485050073, 3.18232319014599673034127887473, 4.29224772394410186057731514337, 5.06969005536045562748249202702, 5.71535285169263474909179632605, 6.34607015899587363027188335136, 7.48856365508286660414044298547, 8.552089167541787589908249447692, 9.419951218644706954839058374670, 9.772434884787074867010224960147

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