Properties

Label 2-5e4-5.4-c1-0-6
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  − 2.08i·2-s + 2.19i·3-s − 2.34·4-s + 4.58·6-s + 0.992i·7-s + 0.726i·8-s − 1.83·9-s + 2·11-s − 5.16i·12-s + 3.37i·13-s + 2.06·14-s − 3.18·16-s + 2.89i·17-s + 3.82i·18-s + 2.58·19-s + ⋯
L(s)  = 1  − 1.47i·2-s + 1.26i·3-s − 1.17·4-s + 1.87·6-s + 0.375i·7-s + 0.256i·8-s − 0.611·9-s + 0.603·11-s − 1.49i·12-s + 0.935i·13-s + 0.553·14-s − 0.795·16-s + 0.702i·17-s + 0.901i·18-s + 0.592·19-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

\(L(1)\) \(\approx\) \(1.44828\)
\(L(\frac12)\) \(\approx\) \(1.44828\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.08iT - 2T^{2} \)
3 \( 1 - 2.19iT - 3T^{2} \)
7 \( 1 - 0.992iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 - 2.89iT - 17T^{2} \)
19 \( 1 - 2.58T + 19T^{2} \)
23 \( 1 - 4.54iT - 23T^{2} \)
29 \( 1 - 5.38T + 29T^{2} \)
31 \( 1 - 0.136T + 31T^{2} \)
37 \( 1 + 2.14iT - 37T^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + 4.64iT - 43T^{2} \)
47 \( 1 - 9.92iT - 47T^{2} \)
53 \( 1 + 7.56iT - 53T^{2} \)
59 \( 1 + 4.91T + 59T^{2} \)
61 \( 1 + 2.76T + 61T^{2} \)
67 \( 1 - 2.18iT - 67T^{2} \)
71 \( 1 - 9.64T + 71T^{2} \)
73 \( 1 + 0.775iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 1.77iT - 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−10.71258913956373070916988698415, −9.720959733632630671937613928362, −9.421848048364219610651371341130, −8.575836866344884047546368753854, −7.03238193690150400286030751513, −5.73015471711992844409056218108, −4.48527994715935859344201098111, −3.93229807316909968410896507126, −2.92866501857315221728458968662, −1.54459962503501652282022184823, 0.891605398386285778221091692255, 2.66344085466365775473730107319, 4.40239656489755064252275269882, 5.53018465512495221968321406194, 6.41883854166076570306138838505, 7.02494590598812800785261729146, 7.72838044515040667764814832692, 8.362061383026194372221297607555, 9.371202112711464115517070381587, 10.55132491165536964485064226631

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