Properties

Label 2-40e2-20.3-c1-0-21
Degree $2$
Conductor $1600$
Sign $0.525 + 0.850i$
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 + i)3-s + (1 + i)7-s + i·9-s − 4i·11-s + (−4 − 4i)13-s + (−4 + 4i)17-s + 4·19-s − 2·21-s + (5 − 5i)23-s + (−4 − 4i)27-s − 2i·29-s − 8i·31-s + (4 + 4i)33-s + 8·39-s − 4·41-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (0.377 + 0.377i)7-s + 0.333i·9-s − 1.20i·11-s + (−1.10 − 1.10i)13-s + (−0.970 + 0.970i)17-s + 0.917·19-s − 0.436·21-s + (1.04 − 1.04i)23-s + (−0.769 − 0.769i)27-s − 0.371i·29-s − 1.43i·31-s + (0.696 + 0.696i)33-s + 1.28·39-s − 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9550078910\)
\(L(\frac12)\) \(\approx\) \(0.9550078910\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + (4 + 4i)T + 13iT^{2} \)
17 \( 1 + (4 - 4i)T - 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-5 + 5i)T - 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + (7 - 7i)T - 43iT^{2} \)
47 \( 1 + (-3 - 3i)T + 47iT^{2} \)
53 \( 1 + (4 + 4i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + 16iT - 71T^{2} \)
73 \( 1 + (-4 - 4i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (-12 + 12i)T - 97iT^{2} \)
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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

−9.403435773382798172488660615074, −8.310488556918893265567780529407, −7.970434541080482739222639776938, −6.73297676475925844216356226049, −5.79225364350964496423268846699, −5.19295934726610047977433885556, −4.46462991134290027235382930180, −3.23976030722810859147189041380, −2.22956301135360237128084143282, −0.42983496778959672616558086028, 1.21720050887824962521273557664, 2.26110268668030789283699228157, 3.62600451083269632399138749568, 4.88516503467094398560240983087, 5.15012851878892312761354036258, 6.74290309768572867376203147843, 6.99585954222057837778936537293, 7.54279911710181256328349125209, 8.977489032846631961492653431021, 9.435330880818362861898421988574

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