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Properties

Label 2-5e2-5.4-c33-0-2
Degree $2$
Conductor $25$
Sign $0.447 - 0.894i$
Analytic cond. $172.457$
Root an. cond. $13.1322$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.32e5i·2-s + 1.17e8i·3-s − 9.07e9·4-s − 1.56e13·6-s − 1.34e13i·7-s − 6.49e13i·8-s − 8.25e15·9-s − 7.96e16·11-s − 1.06e18i·12-s + 1.16e18i·13-s + 1.79e18·14-s − 6.93e19·16-s + 3.27e20i·17-s − 1.09e21i·18-s − 2.21e21·19-s + ⋯
L(s)  = 1  + 1.43i·2-s + 1.57i·3-s − 1.05·4-s − 2.26·6-s − 0.153i·7-s − 0.0815i·8-s − 1.48·9-s − 0.522·11-s − 1.66i·12-s + 0.486i·13-s + 0.219·14-s − 0.939·16-s + 1.63i·17-s − 2.12i·18-s − 1.76·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(172.457\)
Root analytic conductor: \(13.1322\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :33/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(17)\) \(\approx\) \(0.1554568810\)
\(L(\frac12)\) \(\approx\) \(0.1554568810\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.32e5iT - 8.58e9T^{2} \)
3 \( 1 - 1.17e8iT - 5.55e15T^{2} \)
7 \( 1 + 1.34e13iT - 7.73e27T^{2} \)
11 \( 1 + 7.96e16T + 2.32e34T^{2} \)
13 \( 1 - 1.16e18iT - 5.75e36T^{2} \)
17 \( 1 - 3.27e20iT - 4.02e40T^{2} \)
19 \( 1 + 2.21e21T + 1.58e42T^{2} \)
23 \( 1 + 3.93e22iT - 8.65e44T^{2} \)
29 \( 1 + 1.59e24T + 1.81e48T^{2} \)
31 \( 1 - 2.07e24T + 1.64e49T^{2} \)
37 \( 1 + 4.76e25iT - 5.63e51T^{2} \)
41 \( 1 + 6.39e26T + 1.66e53T^{2} \)
43 \( 1 - 2.16e26iT - 8.02e53T^{2} \)
47 \( 1 - 7.45e27iT - 1.51e55T^{2} \)
53 \( 1 + 1.70e28iT - 7.96e56T^{2} \)
59 \( 1 - 5.91e28T + 2.74e58T^{2} \)
61 \( 1 - 2.18e29T + 8.23e58T^{2} \)
67 \( 1 - 2.21e30iT - 1.82e60T^{2} \)
71 \( 1 - 2.87e30T + 1.23e61T^{2} \)
73 \( 1 - 3.41e30iT - 3.08e61T^{2} \)
79 \( 1 + 5.87e30T + 4.18e62T^{2} \)
83 \( 1 - 2.11e31iT - 2.13e63T^{2} \)
89 \( 1 - 2.67e32T + 2.13e64T^{2} \)
97 \( 1 - 1.07e31iT - 3.65e65T^{2} \)
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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

−12.93630398001934286280995887671, −11.02191227523356351282135475844, −10.23953823405647905559802097126, −8.873192635958507949878270032931, −8.199037631157385659233435605216, −6.62614253619166537709054537862, −5.69068246748444769393577832702, −4.56789966866343438627753729988, −3.92797848348813698030270708406, −2.23514204748033604339471587156, 0.03868770458918721846629333644, 0.69264242345870491518276435380, 1.83697122514945304294950225978, 2.40329039385924943791021453324, 3.47752046297065423527398814332, 5.15442806868231077504372258616, 6.60570279137269026781164154159, 7.61629677366772877574166337947, 8.876747193425340039430806824987, 10.23653302195385441688767680683

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