Properties

Label 2-40e2-25.8-c0-0-0
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  + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (−0.309 − 0.0489i)53-s + (−0.363 − 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (−0.309 − 0.0489i)53-s + (−0.363 − 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)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: $\chi_{1600} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2680982497\)
\(L(\frac12)\) \(\approx\) \(0.2680982497\)
\(L(1)\) 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 \( 1 + (0.587 - 0.809i)T \)
good3 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.788527731974733153929059874014, −9.350997963090286875551756062799, −8.217535631359338123247114988506, −7.62584910515695821944884708007, −6.70461946255714347425846606448, −6.15462438214408811283853939027, −4.88325280003660755778165618259, −4.12762036978884540076924463446, −2.96359690191895761035611322186, −2.23430649740132193869229859567, 0.19013571521074047768085060465, 2.11153614819866869293143234843, 3.14431667478551677439917547126, 4.29238463739474150105366897418, 5.20158125612531101956329560133, 5.63780440767012557735951449441, 7.14052166063600575199750817760, 7.57907129566504486210756755169, 8.560697797309119818651870940566, 9.048708383360608740957749958909

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