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Properties

Label 2-50e2-100.39-c0-0-1
Degree $2$
Conductor $2500$
Sign $-0.684 - 0.728i$
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 1.30i)13-s + (−0.809 − 0.587i)16-s + (0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 + 0.587i)29-s − i·32-s + (0.5 + 0.363i)34-s + (−0.809 + 0.587i)36-s + (−0.363 + 0.5i)37-s + (1.30 + 0.951i)41-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 1.30i)13-s + (−0.809 − 0.587i)16-s + (0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 + 0.587i)29-s − i·32-s + (0.5 + 0.363i)34-s + (−0.809 + 0.587i)36-s + (−0.363 + 0.5i)37-s + (1.30 + 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.684 - 0.728i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479248602\)
\(L(\frac12)\) \(\approx\) \(1.479248602\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.496445636298728295464936404242, −8.348672592748943145372782807400, −7.71788385274443859383138602998, −6.99031303148970041669387333234, −6.48976871840374206811010980132, −5.34158576088990507916878257780, −4.73754715452195977521261878263, −4.05790522621463882685411269117, −2.96097112052332881223790415217, −1.82712676092231839045379489486, 0.827590008240657589550111364329, 2.10960781444890083213953858315, 3.11482969979624526931686863808, 3.87972867293138964735737782127, 4.77703150065004177103973924458, 5.54278845859034346296210939074, 6.29355026433041767887133235968, 7.29120994826106365805515275977, 8.009284041744121685008730741988, 9.164732659817823968588371058743

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