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Properties

Label 2-50e2-2500.71-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.0527 - 0.998i$
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  + (−0.137 + 0.990i)2-s + (−0.962 − 0.272i)4-s + (0.823 + 0.567i)5-s + (0.402 − 0.915i)8-s + (0.260 + 0.965i)9-s + (−0.675 + 0.737i)10-s + (0.704 − 1.84i)13-s + (0.850 + 0.525i)16-s + (0.669 + 0.245i)17-s + (−0.992 + 0.125i)18-s + (−0.637 − 0.770i)20-s + (0.356 + 0.934i)25-s + (1.73 + 0.952i)26-s + (−0.435 − 0.185i)29-s + (−0.637 + 0.770i)32-s + ⋯
L(s)  = 1  + (−0.137 + 0.990i)2-s + (−0.962 − 0.272i)4-s + (0.823 + 0.567i)5-s + (0.402 − 0.915i)8-s + (0.260 + 0.965i)9-s + (−0.675 + 0.737i)10-s + (0.704 − 1.84i)13-s + (0.850 + 0.525i)16-s + (0.669 + 0.245i)17-s + (−0.992 + 0.125i)18-s + (−0.637 − 0.770i)20-s + (0.356 + 0.934i)25-s + (1.73 + 0.952i)26-s + (−0.435 − 0.185i)29-s + (−0.637 + 0.770i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0527 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0527 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.0527 - 0.998i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.0527 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273908435\)
\(L(\frac12)\) \(\approx\) \(1.273908435\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.137 - 0.990i)T \)
5 \( 1 + (-0.823 - 0.567i)T \)
good3 \( 1 + (-0.260 - 0.965i)T^{2} \)
7 \( 1 + (-0.535 + 0.844i)T^{2} \)
11 \( 1 + (-0.938 + 0.344i)T^{2} \)
13 \( 1 + (-0.704 + 1.84i)T + (-0.745 - 0.666i)T^{2} \)
17 \( 1 + (-0.669 - 0.245i)T + (0.762 + 0.647i)T^{2} \)
19 \( 1 + (0.778 - 0.627i)T^{2} \)
23 \( 1 + (0.974 - 0.224i)T^{2} \)
29 \( 1 + (0.435 + 0.185i)T + (0.693 + 0.720i)T^{2} \)
31 \( 1 + (-0.762 - 0.647i)T^{2} \)
37 \( 1 + (-1.34 + 0.101i)T + (0.988 - 0.150i)T^{2} \)
41 \( 1 + (0.0738 - 0.0150i)T + (0.920 - 0.391i)T^{2} \)
43 \( 1 + (-0.876 - 0.481i)T^{2} \)
47 \( 1 + (-0.112 + 0.993i)T^{2} \)
53 \( 1 + (0.461 - 0.919i)T + (-0.597 - 0.801i)T^{2} \)
59 \( 1 + (-0.793 + 0.607i)T^{2} \)
61 \( 1 + (1.95 + 0.399i)T + (0.920 + 0.391i)T^{2} \)
67 \( 1 + (-0.899 - 0.437i)T^{2} \)
71 \( 1 + (-0.979 + 0.199i)T^{2} \)
73 \( 1 + (-0.465 - 0.657i)T + (-0.332 + 0.942i)T^{2} \)
79 \( 1 + (-0.260 - 0.965i)T^{2} \)
83 \( 1 + (-0.356 - 0.934i)T^{2} \)
89 \( 1 + (1.07 - 1.44i)T + (-0.285 - 0.958i)T^{2} \)
97 \( 1 + (-0.615 - 1.02i)T + (-0.470 + 0.882i)T^{2} \)
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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

−9.279931942876860779387012107503, −8.196298885434513003423779252677, −7.83669237856981681307112887626, −7.05261277955324782270615956291, −6.01494440423329827613335551340, −5.67024739461686113892527520555, −4.88966937533514460003020965653, −3.70678632278408200248728098057, −2.70914965391438270946664802011, −1.29272071098634559998689315127, 1.15200800345077360613382350984, 1.90310108921681190206678631186, 3.10048007794207035147646303892, 4.09480418625850573168055572999, 4.67237688433843967559208501425, 5.79440329160528522180250276741, 6.44868271951864843192002764162, 7.51075692092561428014922783904, 8.586304875479807573442034267967, 9.196821814296350150346159500165

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