Properties

Label 2-5e2-5.4-c33-0-22
Degree $2$
Conductor $25$
Sign $-0.894 - 0.447i$
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  + 1.77e5i·2-s + 1.20e8i·3-s − 2.27e10·4-s − 2.14e13·6-s + 1.11e14i·7-s − 2.51e15i·8-s − 9.06e15·9-s − 1.24e17·11-s − 2.75e18i·12-s + 2.26e18i·13-s − 1.97e19·14-s + 2.50e20·16-s − 3.98e20i·17-s − 1.60e21i·18-s − 3.39e20·19-s + ⋯
L(s)  = 1  + 1.91i·2-s + 1.62i·3-s − 2.65·4-s − 3.09·6-s + 1.26i·7-s − 3.16i·8-s − 1.62·9-s − 0.819·11-s − 4.30i·12-s + 0.944i·13-s − 2.41·14-s + 3.38·16-s − 1.98i·17-s − 3.11i·18-s − 0.270·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 25,\ (\ :33/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(17)\) \(\approx\) \(0.5745054857\)
\(L(\frac12)\) \(\approx\) \(0.5745054857\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.77e5iT - 8.58e9T^{2} \)
3 \( 1 - 1.20e8iT - 5.55e15T^{2} \)
7 \( 1 - 1.11e14iT - 7.73e27T^{2} \)
11 \( 1 + 1.24e17T + 2.32e34T^{2} \)
13 \( 1 - 2.26e18iT - 5.75e36T^{2} \)
17 \( 1 + 3.98e20iT - 4.02e40T^{2} \)
19 \( 1 + 3.39e20T + 1.58e42T^{2} \)
23 \( 1 - 1.96e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.40e23T + 1.81e48T^{2} \)
31 \( 1 - 3.85e24T + 1.64e49T^{2} \)
37 \( 1 - 5.90e24iT - 5.63e51T^{2} \)
41 \( 1 + 8.25e24T + 1.66e53T^{2} \)
43 \( 1 + 6.30e26iT - 8.02e53T^{2} \)
47 \( 1 + 3.36e27iT - 1.51e55T^{2} \)
53 \( 1 + 5.10e27iT - 7.96e56T^{2} \)
59 \( 1 - 2.91e29T + 2.74e58T^{2} \)
61 \( 1 + 3.91e29T + 8.23e58T^{2} \)
67 \( 1 - 1.32e30iT - 1.82e60T^{2} \)
71 \( 1 + 4.98e30T + 1.23e61T^{2} \)
73 \( 1 - 3.33e30iT - 3.08e61T^{2} \)
79 \( 1 + 1.93e31T + 4.18e62T^{2} \)
83 \( 1 - 6.11e31iT - 2.13e63T^{2} \)
89 \( 1 - 7.66e31T + 2.13e64T^{2} \)
97 \( 1 + 2.66e32iT - 3.65e65T^{2} \)
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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

−11.74086905333329719675711220139, −9.938344753097969649475952424945, −9.194466258418080365274273462848, −8.477669485423291436566395485995, −7.04385516624772832657064958672, −5.64305623494327104959637998875, −5.09868358620337596668048644289, −4.24471184422764464973450686601, −2.83321297586588106337416222281, −0.16556266215886027392592069179, 0.70357936921675538370963026861, 1.33564357192352285684086328097, 2.29051840177087345668432759937, 3.27730733642265675279287816908, 4.48134231221109099290396987782, 6.06800849807409546924794689471, 7.73661146697819270646211174635, 8.422800570650457715383720190808, 10.29028725854509428377311896727, 10.77717977430896916025667444475

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