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Properties

Label 2-5e2-5.4-c13-0-13
Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Analytic cond. $26.8077$
Root an. cond. $5.17761$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 90.6i·2-s − 1.12e3i·3-s − 23.8·4-s + 1.02e5·6-s − 3.24e5i·7-s + 7.40e5i·8-s + 3.26e5·9-s − 1.64e6·11-s + 2.68e4i·12-s + 6.26e6i·13-s + 2.94e7·14-s − 6.73e7·16-s − 1.66e8i·17-s + 2.96e7i·18-s − 3.12e8·19-s + ⋯
L(s)  = 1  + 1.00i·2-s − 0.891i·3-s − 0.00291·4-s + 0.892·6-s − 1.04i·7-s + 0.998i·8-s + 0.205·9-s − 0.280·11-s + 0.00260i·12-s + 0.360i·13-s + 1.04·14-s − 1.00·16-s − 1.67i·17-s + 0.205i·18-s − 1.52·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(26.8077\)
Root analytic conductor: \(5.17761\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :13/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.742101312\)
\(L(\frac12)\) \(\approx\) \(1.742101312\)
\(L(\frac{15}{2})\) 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 \( 1 - 90.6iT - 8.19e3T^{2} \)
3 \( 1 + 1.12e3iT - 1.59e6T^{2} \)
7 \( 1 + 3.24e5iT - 9.68e10T^{2} \)
11 \( 1 + 1.64e6T + 3.45e13T^{2} \)
13 \( 1 - 6.26e6iT - 3.02e14T^{2} \)
17 \( 1 + 1.66e8iT - 9.90e15T^{2} \)
19 \( 1 + 3.12e8T + 4.20e16T^{2} \)
23 \( 1 + 6.32e8iT - 5.04e17T^{2} \)
29 \( 1 - 2.82e9T + 1.02e19T^{2} \)
31 \( 1 - 7.61e9T + 2.44e19T^{2} \)
37 \( 1 + 1.99e10iT - 2.43e20T^{2} \)
41 \( 1 + 4.69e10T + 9.25e20T^{2} \)
43 \( 1 + 7.85e9iT - 1.71e21T^{2} \)
47 \( 1 + 8.31e10iT - 5.46e21T^{2} \)
53 \( 1 + 1.19e11iT - 2.60e22T^{2} \)
59 \( 1 + 4.20e11T + 1.04e23T^{2} \)
61 \( 1 - 4.15e11T + 1.61e23T^{2} \)
67 \( 1 - 1.02e11iT - 5.48e23T^{2} \)
71 \( 1 + 4.00e11T + 1.16e24T^{2} \)
73 \( 1 - 5.55e11iT - 1.67e24T^{2} \)
79 \( 1 + 1.60e12T + 4.66e24T^{2} \)
83 \( 1 + 2.64e11iT - 8.87e24T^{2} \)
89 \( 1 - 3.69e12T + 2.19e25T^{2} \)
97 \( 1 - 1.00e13iT - 6.73e25T^{2} \)
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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

−14.24946515696348169621517654260, −13.37986477140803643377701923253, −11.89960855526678521172773033526, −10.40140152426588236209031423825, −8.360111336210062178806486016013, −7.16463530102650693635804487577, −6.54848126675787261726760229511, −4.64566347049201769966134146010, −2.27110097834501541663473702518, −0.54938783696855979879187211045, 1.64416564588725155655868654872, 3.05464312359247633397441685253, 4.44238941233247622983358453623, 6.24606478013466914800096386632, 8.482245956668591427454111006384, 9.953835685216194709122714068964, 10.70409701219192200708803563064, 12.06914506619936012988633483869, 13.04338908100352549241565241136, 15.21181329336438900704750626340

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