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Properties

Label 2-5e4-25.16-c1-0-25
Degree $2$
Conductor $625$
Sign $0.855 + 0.518i$
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  + (0.855 + 0.621i)2-s + (0.212 + 0.653i)3-s + (−0.272 − 0.838i)4-s + (−0.224 + 0.691i)6-s − 1.01·7-s + (0.941 − 2.89i)8-s + (2.04 − 1.48i)9-s + (−4.14 − 3.00i)11-s + (0.490 − 0.356i)12-s + (4.92 − 3.57i)13-s + (−0.865 − 0.628i)14-s + (1.17 − 0.857i)16-s + (−0.986 + 3.03i)17-s + 2.67·18-s + (1.05 − 3.26i)19-s + ⋯
L(s)  = 1  + (0.604 + 0.439i)2-s + (0.122 + 0.377i)3-s + (−0.136 − 0.419i)4-s + (−0.0916 + 0.282i)6-s − 0.382·7-s + (0.332 − 1.02i)8-s + (0.681 − 0.495i)9-s + (−1.24 − 0.907i)11-s + (0.141 − 0.102i)12-s + (1.36 − 0.992i)13-s + (−0.231 − 0.168i)14-s + (0.294 − 0.214i)16-s + (−0.239 + 0.736i)17-s + 0.629·18-s + (0.243 − 0.748i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86542 - 0.520836i\)
\(L(\frac12)\) \(\approx\) \(1.86542 - 0.520836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.855 - 0.621i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.212 - 0.653i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 1.01T + 7T^{2} \)
11 \( 1 + (4.14 + 3.00i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-4.92 + 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.986 - 3.03i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.05 + 3.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.36 - 1.71i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.479 + 1.47i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.47 - 7.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.80 + 4.94i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.50 + 1.09i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.22T + 43T^{2} \)
47 \( 1 + (-1.48 - 4.56i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.10 + 9.55i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.34 - 1.70i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.86 - 1.35i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.43 - 4.42i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.39 + 7.35i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.481 + 0.350i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.41 - 10.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.41 - 13.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.17 - 3.75i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.35 - 13.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.70617535998141983782162319293, −9.692895335335600827081016186930, −8.855170930726347186315225585887, −7.83313138693949482612992475654, −6.68993812983379337786671691517, −5.87741227408643666516746773896, −5.14818371231554922767395651774, −3.94995952400069235380036082502, −3.12721668473354677869531570416, −0.938101299890577839385381059312, 1.84567072018475272099114708683, 2.87691402034306288849831974507, 4.14414357606373406230537530221, 4.83971536855387736137677316741, 6.11369498520898452749145869441, 7.32950068124539779004157409116, 7.85721361640579013243739788333, 8.920739612637643117846366503673, 9.918687917267692674070846181389, 10.88603170828196088065253536789

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