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Properties

Label 2-5e4-25.6-c1-0-2
Degree $2$
Conductor $625$
Sign $0.268 + 0.963i$
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  + (0.823 + 2.53i)2-s + (−0.614 + 0.446i)3-s + (−4.12 + 2.99i)4-s + (−1.63 − 1.18i)6-s − 2.04·7-s + (−6.68 − 4.85i)8-s + (−0.748 + 2.30i)9-s + (0.416 + 1.28i)11-s + (1.19 − 3.68i)12-s + (0.407 − 1.25i)13-s + (−1.68 − 5.17i)14-s + (3.65 − 11.2i)16-s + (−3.30 − 2.40i)17-s − 6.45·18-s + (3.95 + 2.87i)19-s + ⋯
L(s)  = 1  + (0.582 + 1.79i)2-s + (−0.354 + 0.257i)3-s + (−2.06 + 1.49i)4-s + (−0.668 − 0.485i)6-s − 0.771·7-s + (−2.36 − 1.71i)8-s + (−0.249 + 0.768i)9-s + (0.125 + 0.386i)11-s + (0.345 − 1.06i)12-s + (0.113 − 0.348i)13-s + (−0.449 − 1.38i)14-s + (0.913 − 2.81i)16-s + (−0.801 − 0.582i)17-s − 1.52·18-s + (0.906 + 0.658i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.268 + 0.963i$
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: \((2,\ 625,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525560 - 0.398921i\)
\(L(\frac12)\) \(\approx\) \(0.525560 - 0.398921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.823 - 2.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.614 - 0.446i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 + (-0.416 - 1.28i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.407 + 1.25i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.30 + 2.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.95 - 2.87i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.845 + 2.60i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.73 + 2.71i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.79 + 4.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.67 - 8.22i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.12 - 9.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 + (6.12 - 4.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.502 + 0.365i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.50 - 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.200 - 0.615i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (8.84 + 6.42i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.38 + 1.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.18 - 12.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.52 - 1.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.92 + 1.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.26 - 6.97i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.69 - 3.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.44864456351669691605585404813, −10.10736048596626848900897000188, −9.380345602796340278762126443863, −8.308820719934822331908951841860, −7.63146394695622522502172995774, −6.65924501716394478150536842887, −5.98221164657067265469437927074, −5.06346738164219583857516810947, −4.32814802367962722356375136419, −3.04770224389897542996774609765, 0.31037711411175745129626775785, 1.76325209126700821229393907301, 3.19469815821395295595237707694, 3.75810345461305396936808772418, 5.07029203407572269521709103351, 5.97702706679609746052449695119, 6.96798563199815057176306326365, 8.861592191567568239285872732458, 9.198071685578514194419574123287, 10.19879209308987355992775312297

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