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Properties

Label 2-40e2-200.171-c0-0-0
Degree $2$
Conductor $1600$
Sign $-0.0941 - 0.995i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.0941 - 0.995i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.0941 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2877621363\)
\(L(\frac12)\) \(\approx\) \(0.2877621363\)
\(L(1)\) 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

−9.778692172501007616369896441331, −9.063015252805024902508335255578, −8.137016478047965306776822993471, −7.28786096843789182161683084708, −6.30929949036487027718291607962, −5.92379029379434885826052551522, −5.01640964609491541991965405992, −4.40345633794233570294853193848, −2.18014084547691591021721697618, −1.81438034739428877381872476853, 0.24354205134050874608207996504, 2.36432865045722961634071350977, 3.61810724124966905496197926943, 4.46471119406299501610133576373, 5.34983746784301941266299364124, 5.99510410612448400109295408860, 6.87793297424145560448660259010, 7.46713389173889302612800606721, 8.874141712071997233471167777931, 9.744871370863205168409901409902

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