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Properties

Label 2-50e2-2500.471-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.871 - 0.490i$
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  + (−0.236 − 0.971i)2-s + (−0.888 + 0.459i)4-s + (0.994 + 0.100i)5-s + (0.656 + 0.754i)8-s + (−0.675 + 0.737i)9-s + (−0.137 − 0.990i)10-s + (−1.95 + 0.399i)13-s + (0.577 − 0.816i)16-s + (0.878 + 1.75i)17-s + (0.876 + 0.481i)18-s + (−0.929 + 0.368i)20-s + (0.979 + 0.199i)25-s + (0.851 + 1.80i)26-s + (−1.06 + 1.58i)29-s + (−0.929 − 0.368i)32-s + ⋯
L(s)  = 1  + (−0.236 − 0.971i)2-s + (−0.888 + 0.459i)4-s + (0.994 + 0.100i)5-s + (0.656 + 0.754i)8-s + (−0.675 + 0.737i)9-s + (−0.137 − 0.990i)10-s + (−1.95 + 0.399i)13-s + (0.577 − 0.816i)16-s + (0.878 + 1.75i)17-s + (0.876 + 0.481i)18-s + (−0.929 + 0.368i)20-s + (0.979 + 0.199i)25-s + (0.851 + 1.80i)26-s + (−1.06 + 1.58i)29-s + (−0.929 − 0.368i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.871 - 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8671045749\)
\(L(\frac12)\) \(\approx\) \(0.8671045749\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.236 + 0.971i)T \)
5 \( 1 + (-0.994 - 0.100i)T \)
good3 \( 1 + (0.675 - 0.737i)T^{2} \)
7 \( 1 + (0.637 + 0.770i)T^{2} \)
11 \( 1 + (-0.448 + 0.893i)T^{2} \)
13 \( 1 + (1.95 - 0.399i)T + (0.920 - 0.391i)T^{2} \)
17 \( 1 + (-0.878 - 1.75i)T + (-0.597 + 0.801i)T^{2} \)
19 \( 1 + (-0.112 - 0.993i)T^{2} \)
23 \( 1 + (0.0376 + 0.999i)T^{2} \)
29 \( 1 + (1.06 - 1.58i)T + (-0.379 - 0.925i)T^{2} \)
31 \( 1 + (0.597 - 0.801i)T^{2} \)
37 \( 1 + (0.00346 + 0.275i)T + (-0.999 + 0.0251i)T^{2} \)
41 \( 1 + (-0.669 + 1.25i)T + (-0.556 - 0.830i)T^{2} \)
43 \( 1 + (0.425 + 0.904i)T^{2} \)
47 \( 1 + (-0.693 - 0.720i)T^{2} \)
53 \( 1 + (0.624 - 1.77i)T + (-0.778 - 0.627i)T^{2} \)
59 \( 1 + (-0.402 + 0.915i)T^{2} \)
61 \( 1 + (-0.813 - 1.52i)T + (-0.556 + 0.830i)T^{2} \)
67 \( 1 + (0.997 + 0.0753i)T^{2} \)
71 \( 1 + (0.470 - 0.882i)T^{2} \)
73 \( 1 + (-0.468 - 1.22i)T + (-0.745 + 0.666i)T^{2} \)
79 \( 1 + (0.675 - 0.737i)T^{2} \)
83 \( 1 + (-0.979 - 0.199i)T^{2} \)
89 \( 1 + (-1.55 + 1.25i)T + (0.212 - 0.977i)T^{2} \)
97 \( 1 + (1.46 - 0.536i)T + (0.762 - 0.647i)T^{2} \)
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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

−9.275171581358738202091991776153, −8.672337098090337678145805103873, −7.76141536738252018934216311945, −7.06948845909340651071891774769, −5.68726972044108084710196723124, −5.31079873686686819037488259506, −4.32359894480361767000169854559, −3.18478373111768464810650452518, −2.31178183237049375496110935873, −1.64335649248324375156032704868, 0.60753803385429031847000995135, 2.29922349228816305180444020260, 3.28055905058957139884460132300, 4.77925278237227804760134743538, 5.19344157668565492066261577803, 5.99030599154114417792787193335, 6.69990417077594274301695407255, 7.54454039232271771835695124950, 8.094554041847160232618395643156, 9.278828051025818061404416890625

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