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Properties

Label 2-5e2-5.4-c9-0-10
Degree $2$
Conductor $25$
Sign $-0.894 - 0.447i$
Analytic cond. $12.8758$
Root an. cond. $3.58829$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.4i·2-s − 210. i·3-s + 53.2·4-s − 4.50e3·6-s − 9.90e3i·7-s − 1.21e4i·8-s − 2.44e4·9-s + 3.64e4·11-s − 1.11e4i·12-s + 1.64e5i·13-s − 2.12e5·14-s − 2.32e5·16-s − 8.23e4i·17-s + 5.24e5i·18-s + 6.09e5·19-s + ⋯
L(s)  = 1  − 0.946i·2-s − 1.49i·3-s + 0.103·4-s − 1.41·6-s − 1.55i·7-s − 1.04i·8-s − 1.24·9-s + 0.750·11-s − 0.155i·12-s + 1.60i·13-s − 1.47·14-s − 0.885·16-s − 0.239i·17-s + 1.17i·18-s + 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(12.8758\)
Root analytic conductor: \(3.58829\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :9/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.449240 + 1.90301i\)
\(L(\frac12)\) \(\approx\) \(0.449240 + 1.90301i\)
\(L(\frac{11}{2})\) 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 \( 1 + 21.4iT - 512T^{2} \)
3 \( 1 + 210. iT - 1.96e4T^{2} \)
7 \( 1 + 9.90e3iT - 4.03e7T^{2} \)
11 \( 1 - 3.64e4T + 2.35e9T^{2} \)
13 \( 1 - 1.64e5iT - 1.06e10T^{2} \)
17 \( 1 + 8.23e4iT - 1.18e11T^{2} \)
19 \( 1 - 6.09e5T + 3.22e11T^{2} \)
23 \( 1 - 1.88e6iT - 1.80e12T^{2} \)
29 \( 1 + 3.39e5T + 1.45e13T^{2} \)
31 \( 1 - 5.47e5T + 2.64e13T^{2} \)
37 \( 1 + 5.25e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.05e6T + 3.27e14T^{2} \)
43 \( 1 + 6.76e6iT - 5.02e14T^{2} \)
47 \( 1 + 3.15e7iT - 1.11e15T^{2} \)
53 \( 1 + 4.89e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.77e7T + 8.66e15T^{2} \)
61 \( 1 - 3.84e7T + 1.16e16T^{2} \)
67 \( 1 - 1.36e8iT - 2.72e16T^{2} \)
71 \( 1 - 3.49e8T + 4.58e16T^{2} \)
73 \( 1 - 1.61e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.26e8T + 1.19e17T^{2} \)
83 \( 1 + 2.87e8iT - 1.86e17T^{2} \)
89 \( 1 - 5.63e8T + 3.50e17T^{2} \)
97 \( 1 - 4.71e8iT - 7.60e17T^{2} \)
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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

−14.03098562350195144253718799910, −13.40079210603887891623012334601, −11.99952042668399417528961832397, −11.27607933268304140162693184731, −9.609846603011036650800131458475, −7.37397584050575960425352729585, −6.73314646366840056569185993614, −3.76744448635663106736258238523, −1.77405655959351973659783273008, −0.885837628348138331128736733116, 2.92158998976298554817300613944, 5.03948903960938762892291819319, 6.04068486171003362582004589949, 8.215942398066601298765365963050, 9.360880484798921176871248760091, 10.83742199016992087505907288918, 12.19736800245643693523383370877, 14.50717613511932824686462650206, 15.28937053119848867955621110383, 15.84486935846594716325596010557

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