Properties

Label 2-50e2-100.19-c0-0-0
Degree $2$
Conductor $2500$
Sign $-0.844 - 0.535i$
Analytic cond. $1.24766$
Root an. cond. $1.11698$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.587 − 0.190i)13-s + (0.309 − 0.951i)16-s + (−0.951 + 1.30i)17-s − 0.999i·18-s + 0.618·26-s + (−1.30 + 0.951i)29-s + i·32-s + (0.499 − 1.53i)34-s + (0.309 + 0.951i)36-s + (−1.53 − 0.5i)37-s + (0.190 − 0.587i)41-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−0.587 − 0.190i)13-s + (0.309 − 0.951i)16-s + (−0.951 + 1.30i)17-s − 0.999i·18-s + 0.618·26-s + (−1.30 + 0.951i)29-s + i·32-s + (0.499 − 1.53i)34-s + (0.309 + 0.951i)36-s + (−1.53 − 0.5i)37-s + (0.190 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3721493858\)
\(L(\frac12)\) \(\approx\) \(0.3721493858\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
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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

−9.202563327004416318755911528161, −8.678654596774608049580928189359, −7.935967718288424748632669553073, −7.29001726624829708285607977023, −6.52274568772857517739865075795, −5.60135395759732927501750367208, −4.98435021692167752823488454390, −3.69611484380375774875207772922, −2.42315183133040485919879929671, −1.70052029853804210968613347936, 0.31092466616925168089455889371, 1.85697905770701417153395102395, 2.82982317526351385012324958370, 3.68240626458517311079111153570, 4.77413799802218622767226746387, 5.90550297720124569487612395058, 6.77647710459961808110992484709, 7.24345884711880690043258695895, 8.194087682415182038248532710161, 8.937854603971365743007000525474

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