Properties

Label 2-40e2-100.31-c0-0-2
Degree $2$
Conductor $1600$
Sign $0.968 + 0.248i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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  + (0.809 + 0.587i)5-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (−0.5 − 0.363i)17-s + (0.309 + 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−1.30 + 0.951i)53-s + (0.5 + 1.53i)61-s + (1.30 − 0.951i)65-s + (−0.5 − 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)5-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (−0.5 − 0.363i)17-s + (0.309 + 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−1.30 + 0.951i)53-s + (0.5 + 1.53i)61-s + (1.30 − 0.951i)65-s + (−0.5 − 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317301027\)
\(L(\frac12)\) \(\approx\) \(1.317301027\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.666962531555811282293756592510, −8.876469138694748065824052559627, −7.983798737799424788397465914883, −7.04094524146893325818338596307, −6.28398985426557245456819981092, −5.69520076277469174969051435177, −4.60621590431531897335432868065, −3.38310452449046606068916386727, −2.70502670592038799258822071380, −1.22470948941782794062876310414, 1.60337412170165041262684511696, 2.28758774429350980591967667312, 3.88433363542550711663554815012, 4.70455789732372464393497721490, 5.46764742139204314842131343284, 6.44551046213581650169088745613, 7.10202434112537957882218144565, 8.257171250378293586665876277720, 8.855415774501323226738822145814, 9.544384037904945321267680352639

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