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Properties

Label 2-5e4-25.6-c1-0-27
Degree $2$
Conductor $625$
Sign $0.268 + 0.963i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.100 − 0.310i)2-s + (1.38 − 1.00i)3-s + (1.53 − 1.11i)4-s + (−0.453 − 0.329i)6-s + 3.42·7-s + (−1.02 − 0.747i)8-s + (−0.0177 + 0.0546i)9-s + (−1.65 − 5.07i)11-s + (1.00 − 3.08i)12-s + (−1.08 + 3.34i)13-s + (−0.345 − 1.06i)14-s + (1.04 − 3.20i)16-s + (−2.06 − 1.50i)17-s + 0.0187·18-s + (1.63 + 1.19i)19-s + ⋯
L(s)  = 1  + (−0.0713 − 0.219i)2-s + (0.801 − 0.582i)3-s + (0.765 − 0.556i)4-s + (−0.185 − 0.134i)6-s + 1.29·7-s + (−0.363 − 0.264i)8-s + (−0.00592 + 0.0182i)9-s + (−0.497 − 1.53i)11-s + (0.289 − 0.891i)12-s + (−0.301 + 0.928i)13-s + (−0.0924 − 0.284i)14-s + (0.260 − 0.801i)16-s + (−0.501 − 0.364i)17-s + 0.00442·18-s + (0.375 + 0.273i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85620 - 1.40893i\)
\(L(\frac12)\) \(\approx\) \(1.85620 - 1.40893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.100 + 0.310i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.38 + 1.00i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + (1.65 + 5.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.08 - 3.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.06 + 1.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.63 - 1.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.33 - 7.20i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.83 - 2.78i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.31 + 0.954i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.00414 - 0.0127i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.99 + 9.20i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.32T + 43T^{2} \)
47 \( 1 + (5.61 - 4.08i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.39 - 1.01i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.00685 - 0.0210i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.21 - 3.72i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.33 + 2.42i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (1.89 - 1.37i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.467 - 1.43i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.345 - 0.250i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.88 - 3.55i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.88 + 5.79i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.9 + 9.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.76662390572707943579761692732, −9.382569694227485395124180742832, −8.668748642768981559482519460563, −7.68653033072738624645790944499, −7.18710814720644750647359489228, −5.83777397441497420294194986729, −5.05400360951486731540981839333, −3.37924789073290167035427608223, −2.27634951772168232137489782727, −1.39204024302335514959539392354, 2.05570273484564940375030103514, 2.91238501952028682280269742079, 4.25165373860391062123897057896, 5.06490738655536329653345954632, 6.49111761010388133556126689713, 7.56904937219632493000120002082, 8.052599355953111206311249273366, 8.869813219668289897374458189493, 9.924971730298568800145783569475, 10.72380802578048235001107657187

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