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Properties

Label 2-5e2-5.3-c8-0-10
Degree $2$
Conductor $25$
Sign $-0.793 - 0.608i$
Analytic cond. $10.1844$
Root an. cond. $3.19131$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.1 − 18.1i)2-s + (−95.3 − 95.3i)3-s − 401. i·4-s − 3.45e3·6-s + (1.47e3 − 1.47e3i)7-s + (−2.63e3 − 2.63e3i)8-s + 1.16e4i·9-s − 1.63e4·11-s + (−3.82e4 + 3.82e4i)12-s + (1.17e4 + 1.17e4i)13-s − 5.36e4i·14-s + 7.29e3·16-s + (2.01e4 − 2.01e4i)17-s + (2.10e5 + 2.10e5i)18-s − 1.19e5i·19-s + ⋯
L(s)  = 1  + (1.13 − 1.13i)2-s + (−1.17 − 1.17i)3-s − 1.56i·4-s − 2.66·6-s + (0.615 − 0.615i)7-s + (−0.642 − 0.642i)8-s + 1.77i·9-s − 1.11·11-s + (−1.84 + 1.84i)12-s + (0.410 + 0.410i)13-s − 1.39i·14-s + 0.111·16-s + (0.241 − 0.241i)17-s + (2.00 + 2.00i)18-s − 0.920i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(10.1844\)
Root analytic conductor: \(3.19131\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :4),\ -0.793 - 0.608i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.576986 + 1.70043i\)
\(L(\frac12)\) \(\approx\) \(0.576986 + 1.70043i\)
\(L(5)\) 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 \( 1 + (-18.1 + 18.1i)T - 256iT^{2} \)
3 \( 1 + (95.3 + 95.3i)T + 6.56e3iT^{2} \)
7 \( 1 + (-1.47e3 + 1.47e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 1.63e4T + 2.14e8T^{2} \)
13 \( 1 + (-1.17e4 - 1.17e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-2.01e4 + 2.01e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.19e5iT - 1.69e10T^{2} \)
23 \( 1 + (3.61e5 + 3.61e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 6.49e4iT - 5.00e11T^{2} \)
31 \( 1 + 9.06e5T + 8.52e11T^{2} \)
37 \( 1 + (-1.51e6 + 1.51e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.57e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.04e6 - 2.04e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-1.34e6 + 1.34e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (7.73e5 + 7.73e5i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.56e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.14e6T + 1.91e14T^{2} \)
67 \( 1 + (-2.76e7 + 2.76e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 1.45e7T + 6.45e14T^{2} \)
73 \( 1 + (1.43e7 + 1.43e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 4.59e7iT - 1.51e15T^{2} \)
83 \( 1 + (2.00e7 + 2.00e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 3.53e7iT - 3.93e15T^{2} \)
97 \( 1 + (-5.33e6 + 5.33e6i)T - 7.83e15iT^{2} \)
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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

−14.27193261400934336277243495343, −13.25569429318652994107917550059, −12.44489857065048384944074450822, −11.30308693107051566567875739987, −10.63140185629027411839760212696, −7.59423982767235961586524113499, −5.89162342589435236293036487065, −4.58127604920399233478325815809, −2.18011530763156269302362219803, −0.68243613079548681718392084082, 3.91472807034598414408342037323, 5.33314241559026463594281485834, 5.83923040925442601478521476399, 7.933135782144447398796236055791, 10.11766882707583474989595041703, 11.50141850918082423711913677167, 12.78092060888359920586878412250, 14.41087179744578398284716742752, 15.55681945109098874347309319719, 15.98262876636362738723259020761

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