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Properties

Label 2-5e2-5.4-c33-0-0
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  + 1.00e5i·2-s − 1.18e8i·3-s − 1.46e9·4-s + 1.18e13·6-s + 1.07e13i·7-s + 7.14e14i·8-s − 8.41e15·9-s + 2.38e17·11-s + 1.73e17i·12-s + 1.09e18i·13-s − 1.07e18·14-s − 8.42e19·16-s + 2.82e20i·17-s − 8.43e20i·18-s − 4.67e20·19-s + ⋯
L(s)  = 1  + 1.08i·2-s − 1.58i·3-s − 0.171·4-s + 1.71·6-s + 0.122i·7-s + 0.897i·8-s − 1.51·9-s + 1.56·11-s + 0.271i·12-s + 0.457i·13-s − 0.132·14-s − 1.14·16-s + 1.40i·17-s − 1.63i·18-s − 0.372·19-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

\(L(17)\) \(\approx\) \(0.006449297174\)
\(L(\frac12)\) \(\approx\) \(0.006449297174\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.00e5iT - 8.58e9T^{2} \)
3 \( 1 + 1.18e8iT - 5.55e15T^{2} \)
7 \( 1 - 1.07e13iT - 7.73e27T^{2} \)
11 \( 1 - 2.38e17T + 2.32e34T^{2} \)
13 \( 1 - 1.09e18iT - 5.75e36T^{2} \)
17 \( 1 - 2.82e20iT - 4.02e40T^{2} \)
19 \( 1 + 4.67e20T + 1.58e42T^{2} \)
23 \( 1 - 1.10e21iT - 8.65e44T^{2} \)
29 \( 1 + 1.88e24T + 1.81e48T^{2} \)
31 \( 1 + 7.01e24T + 1.64e49T^{2} \)
37 \( 1 - 6.16e25iT - 5.63e51T^{2} \)
41 \( 1 - 2.46e25T + 1.66e53T^{2} \)
43 \( 1 + 2.92e25iT - 8.02e53T^{2} \)
47 \( 1 + 4.02e27iT - 1.51e55T^{2} \)
53 \( 1 + 4.79e28iT - 7.96e56T^{2} \)
59 \( 1 + 1.22e29T + 2.74e58T^{2} \)
61 \( 1 - 7.09e28T + 8.23e58T^{2} \)
67 \( 1 + 4.28e29iT - 1.82e60T^{2} \)
71 \( 1 - 6.98e30T + 1.23e61T^{2} \)
73 \( 1 + 5.69e30iT - 3.08e61T^{2} \)
79 \( 1 + 2.53e31T + 4.18e62T^{2} \)
83 \( 1 + 6.16e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.29e32T + 2.13e64T^{2} \)
97 \( 1 - 4.34e32iT - 3.65e65T^{2} \)
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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

−12.10727702887458326396087043390, −11.17914613215132605836609928671, −9.031434898549365351876179653109, −8.115602742410151045142939769816, −7.05112464953282272357602897977, −6.48044110606966234852641170716, −5.62173324356591888554805455471, −3.80953171622501542107750274694, −2.00050107427604830345006820889, −1.53836483124323515454263076829, 0.00110872953366768941620500713, 1.23812386892200480986681802655, 2.60165142693177709832609493751, 3.68140075046246991815110108882, 4.20249730688819764489010315859, 5.57586607351163500221418901209, 7.11705919518594003299640295717, 9.199927351412251631370824954458, 9.507140691297542535370750293298, 10.80381960496155159308568886547

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