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Properties

Label 2-5e2-5.4-c11-0-5
Degree $2$
Conductor $25$
Sign $-0.447 - 0.894i$
Analytic cond. $19.2085$
Root an. cond. $4.38275$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24i·2-s + 252i·3-s + 1.47e3·4-s − 6.04e3·6-s + 1.67e4i·7-s + 8.44e4i·8-s + 1.13e5·9-s + 5.34e5·11-s + 3.70e5i·12-s − 5.77e5i·13-s − 4.01e5·14-s + 9.87e5·16-s + 6.90e6i·17-s + 2.72e6i·18-s − 1.06e7·19-s + ⋯
L(s)  = 1  + 0.530i·2-s + 0.598i·3-s + 0.718·4-s − 0.317·6-s + 0.376i·7-s + 0.911i·8-s + 0.641·9-s + 1.00·11-s + 0.430i·12-s − 0.431i·13-s − 0.199·14-s + 0.235·16-s + 1.17i·17-s + 0.340i·18-s − 0.987·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(19.2085\)
Root analytic conductor: \(4.38275\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :11/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.27076 + 2.05614i\)
\(L(\frac12)\) \(\approx\) \(1.27076 + 2.05614i\)
\(L(\frac{13}{2})\) 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 \( 1 - 24iT - 2.04e3T^{2} \)
3 \( 1 - 252iT - 1.77e5T^{2} \)
7 \( 1 - 1.67e4iT - 1.97e9T^{2} \)
11 \( 1 - 5.34e5T + 2.85e11T^{2} \)
13 \( 1 + 5.77e5iT - 1.79e12T^{2} \)
17 \( 1 - 6.90e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.06e7T + 1.16e14T^{2} \)
23 \( 1 - 1.86e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.28e8T + 1.22e16T^{2} \)
31 \( 1 + 5.28e7T + 2.54e16T^{2} \)
37 \( 1 - 1.82e8iT - 1.77e17T^{2} \)
41 \( 1 - 3.08e8T + 5.50e17T^{2} \)
43 \( 1 + 1.71e7iT - 9.29e17T^{2} \)
47 \( 1 + 2.68e9iT - 2.47e18T^{2} \)
53 \( 1 + 1.59e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.18e9T + 3.01e19T^{2} \)
61 \( 1 - 6.95e9T + 4.35e19T^{2} \)
67 \( 1 - 1.54e10iT - 1.22e20T^{2} \)
71 \( 1 - 9.79e9T + 2.31e20T^{2} \)
73 \( 1 - 1.46e9iT - 3.13e20T^{2} \)
79 \( 1 + 3.81e10T + 7.47e20T^{2} \)
83 \( 1 + 2.93e10iT - 1.28e21T^{2} \)
89 \( 1 - 2.49e10T + 2.77e21T^{2} \)
97 \( 1 + 7.50e10iT - 7.15e21T^{2} \)
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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

−15.29855715295858882746184393866, −14.75608454664552040295093498462, −12.82177103312229604885240164752, −11.41590413706798709834114318567, −10.17759706355169179010704376730, −8.605203624331552523342271162593, −7.02200768285304446621713590224, −5.68996044964596848995831986223, −3.84792282123345933351020564158, −1.80473776267650033488740826688, 0.926385349266899633125817506920, 2.18443434866471447501051418763, 4.04256954016969285500285515474, 6.48042515491736724143788439794, 7.39440585809827607424925747760, 9.431828185380058711508647246190, 10.87452484069808015158852025020, 12.00134958167696348120128666180, 13.02665133579539324773727066782, 14.46103779284686886134617321896

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