Properties

Label 2-50e2-2500.1211-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.983 - 0.179i$
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  + (−0.999 + 0.0251i)2-s + (0.998 − 0.0502i)4-s + (−0.984 − 0.175i)5-s + (−0.997 + 0.0753i)8-s + (0.617 + 0.786i)9-s + (0.988 + 0.150i)10-s + (−1.26 + 0.465i)13-s + (0.994 − 0.100i)16-s + (0.669 − 1.75i)17-s + (−0.637 − 0.770i)18-s + (−0.992 − 0.125i)20-s + (0.938 + 0.344i)25-s + (1.25 − 0.497i)26-s + (1.03 − 1.38i)29-s + (−0.992 + 0.125i)32-s + ⋯
L(s)  = 1  + (−0.999 + 0.0251i)2-s + (0.998 − 0.0502i)4-s + (−0.984 − 0.175i)5-s + (−0.997 + 0.0753i)8-s + (0.617 + 0.786i)9-s + (0.988 + 0.150i)10-s + (−1.26 + 0.465i)13-s + (0.994 − 0.100i)16-s + (0.669 − 1.75i)17-s + (−0.637 − 0.770i)18-s + (−0.992 − 0.125i)20-s + (0.938 + 0.344i)25-s + (1.25 − 0.497i)26-s + (1.03 − 1.38i)29-s + (−0.992 + 0.125i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.983 - 0.179i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6302894504\)
\(L(\frac12)\) \(\approx\) \(0.6302894504\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.999 - 0.0251i)T \)
5 \( 1 + (0.984 + 0.175i)T \)
good3 \( 1 + (-0.617 - 0.786i)T^{2} \)
7 \( 1 + (-0.728 - 0.684i)T^{2} \)
11 \( 1 + (-0.356 - 0.934i)T^{2} \)
13 \( 1 + (1.26 - 0.465i)T + (0.762 - 0.647i)T^{2} \)
17 \( 1 + (-0.669 + 1.75i)T + (-0.745 - 0.666i)T^{2} \)
19 \( 1 + (0.556 + 0.830i)T^{2} \)
23 \( 1 + (0.947 + 0.320i)T^{2} \)
29 \( 1 + (-1.03 + 1.38i)T + (-0.285 - 0.958i)T^{2} \)
31 \( 1 + (0.745 + 0.666i)T^{2} \)
37 \( 1 + (-0.796 - 1.80i)T + (-0.675 + 0.737i)T^{2} \)
41 \( 1 + (0.751 - 1.49i)T + (-0.597 - 0.801i)T^{2} \)
43 \( 1 + (0.929 - 0.368i)T^{2} \)
47 \( 1 + (-0.162 - 0.986i)T^{2} \)
53 \( 1 + (-1.13 - 0.230i)T + (0.920 + 0.391i)T^{2} \)
59 \( 1 + (-0.899 + 0.437i)T^{2} \)
61 \( 1 + (0.123 + 0.246i)T + (-0.597 + 0.801i)T^{2} \)
67 \( 1 + (-0.793 + 0.607i)T^{2} \)
71 \( 1 + (-0.448 + 0.893i)T^{2} \)
73 \( 1 + (-1.02 + 1.71i)T + (-0.470 - 0.882i)T^{2} \)
79 \( 1 + (-0.617 - 0.786i)T^{2} \)
83 \( 1 + (-0.938 - 0.344i)T^{2} \)
89 \( 1 + (-1.46 + 0.621i)T + (0.693 - 0.720i)T^{2} \)
97 \( 1 + (-1.06 - 1.50i)T + (-0.332 + 0.942i)T^{2} \)
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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

−9.200669230448652525528265203580, −8.153276424516962239235533651946, −7.70521751303507671288139967413, −7.17729222176902211226198600479, −6.38368868047989684204023854498, −4.99379463759317230862917363192, −4.56108512409063788002851175413, −3.12020193827313108376995288023, −2.34922242274688630642679078770, −0.904749205296848276271922105487, 0.825867177455509318480635427478, 2.19359585309151977560355554164, 3.37465449848357747081849261281, 3.98804867971367435350582984068, 5.28328081193252457253903570190, 6.26446093127224200793645510661, 7.19552136533274731089425880163, 7.43723816270775072019252073479, 8.441822626445383563739017014792, 8.882665337702589464365078282280

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