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Properties

Label 2-5e2-5.2-c8-0-8
Degree $2$
Conductor $25$
Sign $0.850 + 0.525i$
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  + (11.8 + 11.8i)2-s + (90.9 − 90.9i)3-s + 25.5i·4-s + 2.15e3·6-s + (−508. − 508. i)7-s + (2.73e3 − 2.73e3i)8-s − 9.98e3i·9-s − 7.02e3·11-s + (2.32e3 + 2.32e3i)12-s + (8.07e3 − 8.07e3i)13-s − 1.20e4i·14-s + 7.14e4·16-s + (1.02e5 + 1.02e5i)17-s + (1.18e5 − 1.18e5i)18-s + 5.95e4i·19-s + ⋯
L(s)  = 1  + (0.741 + 0.741i)2-s + (1.12 − 1.12i)3-s + 0.0998i·4-s + 1.66·6-s + (−0.211 − 0.211i)7-s + (0.667 − 0.667i)8-s − 1.52i·9-s − 0.479·11-s + (0.112 + 0.112i)12-s + (0.282 − 0.282i)13-s − 0.313i·14-s + 1.08·16-s + (1.23 + 1.23i)17-s + (1.12 − 1.12i)18-s + 0.457i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(10.1844\)
Root analytic conductor: \(3.19131\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :4),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.39769 - 0.965213i\)
\(L(\frac12)\) \(\approx\) \(3.39769 - 0.965213i\)
\(L(5)\) 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 \( 1 + (-11.8 - 11.8i)T + 256iT^{2} \)
3 \( 1 + (-90.9 + 90.9i)T - 6.56e3iT^{2} \)
7 \( 1 + (508. + 508. i)T + 5.76e6iT^{2} \)
11 \( 1 + 7.02e3T + 2.14e8T^{2} \)
13 \( 1 + (-8.07e3 + 8.07e3i)T - 8.15e8iT^{2} \)
17 \( 1 + (-1.02e5 - 1.02e5i)T + 6.97e9iT^{2} \)
19 \( 1 - 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.32e5 - 1.32e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 3.92e5iT - 5.00e11T^{2} \)
31 \( 1 + 5.07e5T + 8.52e11T^{2} \)
37 \( 1 + (-6.10e4 - 6.10e4i)T + 3.51e12iT^{2} \)
41 \( 1 + 1.81e6T + 7.98e12T^{2} \)
43 \( 1 + (1.47e6 - 1.47e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-5.66e6 + 5.66e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.74e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.96e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.12e7 - 1.12e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 3.01e7T + 6.45e14T^{2} \)
73 \( 1 + (2.52e7 - 2.52e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 8.14e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.99e7 + 1.99e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 8.20e7iT - 3.93e15T^{2} \)
97 \( 1 + (3.37e7 + 3.37e7i)T + 7.83e15iT^{2} \)
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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

−15.15741384407749440325984403208, −14.30995732060782102689203811951, −13.35548090004284084990500815066, −12.53552510959301641129382448098, −10.16437339172711897835800787569, −8.246167785946133974787713042450, −7.22366833412075383497021408124, −5.80956434209228982078798495343, −3.53046091353710594878284194523, −1.44843683268702691593420488969, 2.54283031587398519213770408769, 3.61964059074509604435871284974, 4.98822930720075372070029357981, 7.917149171669994696191624783498, 9.340065338270996428781770815989, 10.58234616920423578253081868555, 12.04786782485659067530725247434, 13.53657743035946161755380979751, 14.33621324265523046103580106329, 15.61705756857221099292963050425

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