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Properties

Label 2-5e2-5.4-c9-0-4
Degree $2$
Conductor $25$
Sign $0.447 - 0.894i$
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  − 8i·2-s + 114i·3-s + 448·4-s + 912·6-s + 4.24e3i·7-s − 7.68e3i·8-s + 6.68e3·9-s − 4.62e4·11-s + 5.10e4i·12-s + 1.15e5i·13-s + 3.39e4·14-s + 1.67e5·16-s + 4.94e5i·17-s − 5.34e4i·18-s + 1.00e6·19-s + ⋯
L(s)  = 1  − 0.353i·2-s + 0.812i·3-s + 0.875·4-s + 0.287·6-s + 0.667i·7-s − 0.662i·8-s + 0.339·9-s − 0.951·11-s + 0.710i·12-s + 1.12i·13-s + 0.236·14-s + 0.640·16-s + 1.43i·17-s − 0.120i·18-s + 1.77·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(12.8758\)
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: \((2,\ 25,\ (\ :9/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.82494 + 1.12788i\)
\(L(\frac12)\) \(\approx\) \(1.82494 + 1.12788i\)
\(L(\frac{11}{2})\) 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 \( 1 + 8iT - 512T^{2} \)
3 \( 1 - 114iT - 1.96e4T^{2} \)
7 \( 1 - 4.24e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.62e4T + 2.35e9T^{2} \)
13 \( 1 - 1.15e5iT - 1.06e10T^{2} \)
17 \( 1 - 4.94e5iT - 1.18e11T^{2} \)
19 \( 1 - 1.00e6T + 3.22e11T^{2} \)
23 \( 1 - 5.32e5iT - 1.80e12T^{2} \)
29 \( 1 + 4.19e6T + 1.45e13T^{2} \)
31 \( 1 + 3.36e6T + 2.64e13T^{2} \)
37 \( 1 + 1.49e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.10e7T + 3.27e14T^{2} \)
43 \( 1 - 6.39e6iT - 5.02e14T^{2} \)
47 \( 1 + 3.55e7iT - 1.11e15T^{2} \)
53 \( 1 + 3.97e7iT - 3.29e15T^{2} \)
59 \( 1 - 8.51e7T + 8.66e15T^{2} \)
61 \( 1 - 4.57e7T + 1.16e16T^{2} \)
67 \( 1 + 4.52e7iT - 2.72e16T^{2} \)
71 \( 1 + 1.89e8T + 4.58e16T^{2} \)
73 \( 1 + 4.12e8iT - 5.88e16T^{2} \)
79 \( 1 + 9.50e7T + 1.19e17T^{2} \)
83 \( 1 + 2.61e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.99e7T + 3.50e17T^{2} \)
97 \( 1 + 1.95e7iT - 7.60e17T^{2} \)
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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

−15.78543726481236420532012830190, −14.81353919536456208711925480577, −12.92781963465491192705091219246, −11.63363158342774976541999988269, −10.51076079669956257472503771532, −9.338219174520221660525468675806, −7.39235729187329437434913479409, −5.58060200542851640227041726615, −3.65359882793067861220445820694, −1.89821315518625510633248891751, 0.954598983006615405882800477889, 2.79485294626219371182620123805, 5.45056437636152025172754765625, 7.19572065102375042778024019768, 7.71319036573704359358537929929, 10.09157375767329676979363775729, 11.44469568079794094733153013072, 12.79592969474406592692231168612, 13.90394908824848319641188278895, 15.48850309706122149045382192929

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