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Properties

Label 2-5e2-5.4-c33-0-14
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  − 1.82e4i·2-s + 3.67e7i·3-s + 8.25e9·4-s + 6.70e11·6-s − 1.51e13i·7-s − 3.07e14i·8-s + 4.20e15·9-s − 2.61e17·11-s + 3.03e17i·12-s + 3.70e17i·13-s − 2.76e17·14-s + 6.53e19·16-s − 3.71e20i·17-s − 7.68e19i·18-s − 1.42e21·19-s + ⋯
L(s)  = 1  − 0.197i·2-s + 0.492i·3-s + 0.961·4-s + 0.0970·6-s − 0.172i·7-s − 0.386i·8-s + 0.757·9-s − 1.71·11-s + 0.473i·12-s + 0.154i·13-s − 0.0339·14-s + 0.885·16-s − 1.85i·17-s − 0.149i·18-s − 1.13·19-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

\(L(17)\) \(\approx\) \(2.351067262\)
\(L(\frac12)\) \(\approx\) \(2.351067262\)
\(L(\frac{35}{2})\) 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 \( 1 + 1.82e4iT - 8.58e9T^{2} \)
3 \( 1 - 3.67e7iT - 5.55e15T^{2} \)
7 \( 1 + 1.51e13iT - 7.73e27T^{2} \)
11 \( 1 + 2.61e17T + 2.32e34T^{2} \)
13 \( 1 - 3.70e17iT - 5.75e36T^{2} \)
17 \( 1 + 3.71e20iT - 4.02e40T^{2} \)
19 \( 1 + 1.42e21T + 1.58e42T^{2} \)
23 \( 1 - 3.26e22iT - 8.65e44T^{2} \)
29 \( 1 + 6.41e23T + 1.81e48T^{2} \)
31 \( 1 + 1.44e23T + 1.64e49T^{2} \)
37 \( 1 - 1.34e26iT - 5.63e51T^{2} \)
41 \( 1 - 6.37e26T + 1.66e53T^{2} \)
43 \( 1 + 3.37e26iT - 8.02e53T^{2} \)
47 \( 1 - 3.12e27iT - 1.51e55T^{2} \)
53 \( 1 - 4.06e28iT - 7.96e56T^{2} \)
59 \( 1 - 8.77e28T + 2.74e58T^{2} \)
61 \( 1 + 1.84e28T + 8.23e58T^{2} \)
67 \( 1 - 7.69e29iT - 1.82e60T^{2} \)
71 \( 1 - 2.72e30T + 1.23e61T^{2} \)
73 \( 1 + 4.44e30iT - 3.08e61T^{2} \)
79 \( 1 - 1.16e31T + 4.18e62T^{2} \)
83 \( 1 - 4.84e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.18e32T + 2.13e64T^{2} \)
97 \( 1 + 2.71e31iT - 3.65e65T^{2} \)
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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

−11.26098286883180172114558723789, −10.44537907562767273870896710708, −9.519152596400126870536759865828, −7.77433627010861506131763894815, −7.02363439575900582732702836769, −5.56057458411127751188726964971, −4.45802200376022608896813396494, −3.07188432503583587156760014192, −2.23095275252770717189151493754, −0.922407971207842133292003437468, 0.45357142848253631996093133588, 1.91358164485269753717791077145, 2.42784486307895581913151133398, 4.01054478804020566836204802694, 5.55512141917615913114761367192, 6.48688264400321165991913597139, 7.58982788660344833436748391601, 8.357886747343838839807706095498, 10.33156700330633895827374410537, 10.86161778338240107546794127209

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