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Properties

Label 2-5e4-25.6-c1-0-17
Degree $2$
Conductor $625$
Sign $0.699 - 0.714i$
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  + (0.326 + 1.00i)2-s + (0.556 − 0.404i)3-s + (0.713 − 0.518i)4-s + (0.588 + 0.427i)6-s + 1.01·7-s + (2.46 + 1.79i)8-s + (−0.781 + 2.40i)9-s + (1.58 + 4.87i)11-s + (0.187 − 0.576i)12-s + (1.88 − 5.78i)13-s + (0.330 + 1.01i)14-s + (−0.450 + 1.38i)16-s + (−2.58 − 1.87i)17-s − 2.67·18-s + (−2.77 − 2.01i)19-s + ⋯
L(s)  = 1  + (0.231 + 0.711i)2-s + (0.321 − 0.233i)3-s + (0.356 − 0.259i)4-s + (0.240 + 0.174i)6-s + 0.382·7-s + (0.871 + 0.633i)8-s + (−0.260 + 0.801i)9-s + (0.477 + 1.46i)11-s + (0.0540 − 0.166i)12-s + (0.521 − 1.60i)13-s + (0.0883 + 0.271i)14-s + (−0.112 + 0.346i)16-s + (−0.626 − 0.455i)17-s − 0.629·18-s + (−0.636 − 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.699 - 0.714i$
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: \((2,\ 625,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10000 + 0.882761i\)
\(L(\frac12)\) \(\approx\) \(2.10000 + 0.882761i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.326 - 1.00i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.556 + 0.404i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.01T + 7T^{2} \)
11 \( 1 + (-1.58 - 4.87i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.88 + 5.78i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.58 + 1.87i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.77 + 2.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.902 - 2.77i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.25 + 0.912i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.46 - 4.70i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.59 + 7.99i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.575 - 1.77i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.22T + 43T^{2} \)
47 \( 1 + (-3.88 + 2.82i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.13 - 5.90i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.894 + 2.75i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.713 + 2.19i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.76 + 2.73i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-6.25 + 4.54i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.184 + 0.566i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.94 - 6.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.5 + 8.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.97 + 6.08i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-11.3 + 8.27i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.74012023322459078794618189242, −9.944574247894073345048529327813, −8.676090740933906045525920298339, −7.85775316420897096182348370002, −7.23318357910295989997039602163, −6.31459382444132003692177316034, −5.21583863567669057104183432533, −4.55574246430958768800952821481, −2.76367047676488121929848616395, −1.66823920689318520943187044406, 1.38853960966935329077959728865, 2.74083331801455918760857785668, 3.78682233327444163255681354532, 4.41045416954299564376065945685, 6.30466379692469247175026767598, 6.58366721354596117007727461018, 8.227773998228474604791398976688, 8.680737855269693832384032625973, 9.702450726016727919652235854000, 10.75128811641441995984899511934

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