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Properties

Label 2-5e4-25.21-c1-0-9
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  + (−0.622 + 1.91i)2-s + (−2.44 − 1.77i)3-s + (−1.66 − 1.20i)4-s + (4.92 − 3.58i)6-s + 0.369·7-s + (0.0893 − 0.0649i)8-s + (1.90 + 5.85i)9-s + (−0.539 + 1.66i)11-s + (1.92 + 5.91i)12-s + (0.344 + 1.06i)13-s + (−0.230 + 0.708i)14-s + (−1.20 − 3.69i)16-s + (−4.43 + 3.22i)17-s − 12.3·18-s + (3.03 − 2.20i)19-s + ⋯
L(s)  = 1  + (−0.440 + 1.35i)2-s + (−1.41 − 1.02i)3-s + (−0.831 − 0.603i)4-s + (2.01 − 1.46i)6-s + 0.139·7-s + (0.0316 − 0.0229i)8-s + (0.633 + 1.95i)9-s + (−0.162 + 0.500i)11-s + (0.554 + 1.70i)12-s + (0.0956 + 0.294i)13-s + (−0.0615 + 0.189i)14-s + (−0.300 − 0.924i)16-s + (−1.07 + 0.782i)17-s − 2.92·18-s + (0.696 − 0.505i)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: $\chi_{625} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331919 - 0.139526i\)
\(L(\frac12)\) \(\approx\) \(0.331919 - 0.139526i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.622 - 1.91i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.44 + 1.77i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.369T + 7T^{2} \)
11 \( 1 + (0.539 - 1.66i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.344 - 1.06i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.43 - 3.22i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-3.03 + 2.20i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.23 + 6.89i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.39 + 2.46i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.247 - 0.179i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.84 + 8.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.29 + 3.98i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.17T + 43T^{2} \)
47 \( 1 + (0.655 + 0.476i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.16 - 2.30i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.573 + 1.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.99 + 9.21i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-10.0 + 7.32i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (9.44 + 6.86i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.06 + 3.26i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.60 + 3.34i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.77 - 4.19i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.472 + 1.45i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.17 + 3.76i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.70750462025258358976267294143, −9.370570713577775016301531847840, −8.418819400648195531115308821145, −7.47294992220132894051312220848, −6.89116852516527522527326742260, −6.25048521196053520507207074323, −5.43114202953890611641243954698, −4.55046866882729311492945682488, −2.10664465968580592425140058333, −0.31235056381663835784401326138, 1.17380274598307935255528816133, 3.02835608589649836459752221561, 3.99963885829380006849777090803, 5.08765186835988051408230808269, 5.88704151559923961225892413454, 7.04782018415620810466832196885, 8.597445246186477227543267488883, 9.537823374004141718479179232534, 10.00482095896693793475709549526, 10.89124650946669569116034763105

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