Properties

Label 2-40e2-16.5-c1-0-26
Degree $2$
Conductor $1600$
Sign $0.352 + 0.935i$
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  + (0.209 − 0.209i)3-s − 1.73i·7-s + 2.91i·9-s + (−0.505 − 0.505i)11-s + (1.88 − 1.88i)13-s − 4.53·17-s + (3.22 − 3.22i)19-s + (−0.364 − 0.364i)21-s − 8.85i·23-s + (1.23 + 1.23i)27-s + (−2.44 + 2.44i)29-s + 5.70·31-s − 0.211·33-s + (5.35 + 5.35i)37-s − 0.791i·39-s + ⋯
L(s)  = 1  + (0.120 − 0.120i)3-s − 0.656i·7-s + 0.970i·9-s + (−0.152 − 0.152i)11-s + (0.523 − 0.523i)13-s − 1.09·17-s + (0.738 − 0.738i)19-s + (−0.0794 − 0.0794i)21-s − 1.84i·23-s + (0.238 + 0.238i)27-s + (−0.453 + 0.453i)29-s + 1.02·31-s − 0.0368·33-s + (0.880 + 0.880i)37-s − 0.126i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585506442\)
\(L(\frac12)\) \(\approx\) \(1.585506442\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.209 + 0.209i)T - 3iT^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + (0.505 + 0.505i)T + 11iT^{2} \)
13 \( 1 + (-1.88 + 1.88i)T - 13iT^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + (-3.22 + 3.22i)T - 19iT^{2} \)
23 \( 1 + 8.85iT - 23T^{2} \)
29 \( 1 + (2.44 - 2.44i)T - 29iT^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (2.10 + 2.10i)T + 43iT^{2} \)
47 \( 1 - 4.32T + 47T^{2} \)
53 \( 1 + (-1.37 - 1.37i)T + 53iT^{2} \)
59 \( 1 + (6.64 + 6.64i)T + 59iT^{2} \)
61 \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \)
67 \( 1 + (10.5 - 10.5i)T - 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + 6.63iT - 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 + (-9.15 + 9.15i)T - 83iT^{2} \)
89 \( 1 + 3.23iT - 89T^{2} \)
97 \( 1 + 1.94T + 97T^{2} \)
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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

−9.076545061667461098599069293913, −8.466832309309860706460449024152, −7.66319624842674715005909720656, −6.93308038546577937537748357944, −6.08237218855534384878973303914, −4.96222011510499641232784822383, −4.35148633595140799082771656250, −3.10135748950688035322626498622, −2.16315193470187288006134555277, −0.65294235218499124810002173545, 1.31494039629517062340331993230, 2.59194015843742343510107465704, 3.63249621014968770658785474573, 4.43488209812067772511394785082, 5.66079164271156951819511036086, 6.19261978174283679952639244913, 7.16551481931912212269675767606, 8.029682963657187673678584013182, 8.959084341675952543259084730725, 9.424292175376866329529352397383

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