Properties

Label 2-40e2-200.131-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.995 + 0.0941i$
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  + (−1.30 + 0.951i)3-s + (−0.587 − 0.809i)5-s + i·7-s + (0.500 − 1.53i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (1.53 + 0.5i)15-s + (−1.30 − 0.951i)17-s + (−0.5 − 0.363i)19-s + (−0.951 − 1.30i)21-s + (0.951 − 0.309i)23-s + (−0.309 + 0.951i)25-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.587 − 0.809i)31-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)3-s + (−0.587 − 0.809i)5-s + i·7-s + (0.500 − 1.53i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (1.53 + 0.5i)15-s + (−1.30 − 0.951i)17-s + (−0.5 − 0.363i)19-s + (−0.951 − 1.30i)21-s + (0.951 − 0.309i)23-s + (−0.309 + 0.951i)25-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.587 − 0.809i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.995 + 0.0941i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.995 + 0.0941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5785399692\)
\(L(\frac12)\) \(\approx\) \(0.5785399692\)
\(L(1)\) 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 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
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

−9.405011392696965200288382104280, −8.915862771068991011644033077321, −8.349320590885608509007183405913, −6.94107711635354704965862489687, −6.06139093719070633614564303696, −5.43645417122632358300967729752, −4.67770954302149974774492513223, −4.03708820793113223992940035761, −2.70670931210077161827627223170, −0.69715855321432090434568462940, 1.04074018930617883796273314645, 2.37938867806172167380399263916, 3.89629286210148279445147887167, 4.57729930043537049581319044659, 5.81456939005521176085327271372, 6.60431649807921335893372775602, 6.97311726755182227979998665742, 7.74857394998535672768343501178, 8.515126702346343955746390319966, 10.02357230254886395754964909354

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