Properties

Label 2-5e4-5.4-c1-0-28
Degree $2$
Conductor $625$
Sign $-1$
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  − 1.61i·2-s i·3-s − 0.618·4-s − 1.61·6-s + 0.618i·7-s − 2.23i·8-s + 2·9-s − 5.23·11-s + 0.618i·12-s − 1.85i·13-s + 1.00·14-s − 4.85·16-s − 5.23i·17-s − 3.23i·18-s − 0.854·19-s + ⋯
L(s)  = 1  − 1.14i·2-s − 0.577i·3-s − 0.309·4-s − 0.660·6-s + 0.233i·7-s − 0.790i·8-s + 0.666·9-s − 1.57·11-s + 0.178i·12-s − 0.514i·13-s + 0.267·14-s − 1.21·16-s − 1.26i·17-s − 0.762i·18-s − 0.195·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-1$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (624, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35345i\)
\(L(\frac12)\) \(\approx\) \(1.35345i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61iT - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 0.618iT - 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 + 1.85iT - 13T^{2} \)
17 \( 1 + 5.23iT - 17T^{2} \)
19 \( 1 + 0.854T + 19T^{2} \)
23 \( 1 + 3.76iT - 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 0.236iT - 37T^{2} \)
41 \( 1 + 0.763T + 41T^{2} \)
43 \( 1 - 4.85iT - 43T^{2} \)
47 \( 1 - 0.618iT - 47T^{2} \)
53 \( 1 - 3.47iT - 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 4.76iT - 67T^{2} \)
71 \( 1 + 6.61T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 - 6.23iT - 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + 3.85iT - 97T^{2} \)
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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

−10.29335548209582714484894782098, −9.699380988510999225722706772579, −8.459543414156430657438582226724, −7.48428064120398438121225915356, −6.77031415610353377858619929978, −5.44936856000700956702498363574, −4.36563928352683698584166063216, −2.92062907902905855595963599408, −2.25738003964280298841056092760, −0.72262107937754637892936497162, 2.11471709929814528298857042533, 3.73755592235147528671333569411, 4.85306140926171489508089612386, 5.59317323716215296973442642715, 6.65717171646911595805736848425, 7.49007802110720543897693615563, 8.191902905577469752040421952913, 9.126967961139171238911937529022, 10.32326723465647907036878606654, 10.66210096009775161821181164749

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