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Properties

Label 2-40e2-200.131-c0-0-3
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: $\chi_{1600} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.0941 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.563590223\)
\(L(\frac12)\) \(\approx\) \(1.563590223\)
\(L(1)\) 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

−9.120571419011543991382563918680, −8.542619756981637579941484299460, −7.83382670207470998862072145338, −7.09717224697247416514421885225, −6.67824840455401609484910425207, −5.06632009495822976674087939603, −4.07533061721839503246788145211, −3.49302110174771449763827792115, −2.07990678878757648559283531012, −1.18415819880667281241119596464, 2.22609653196111025837970414060, 2.99870716620426868469770204119, 3.78026945593509392405392727301, 4.42380778987143864883022575030, 5.88026000274794413810463069025, 6.46782764380304626273716062707, 7.87431308798989923379696995959, 8.361168877379002630428815869415, 8.870633455126364833599992329547, 9.680676970395542032535209539895

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