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Properties

Label 2-5e2-5.4-c33-0-26
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  − 1.40e5i·2-s − 1.29e8i·3-s − 1.11e10·4-s − 1.81e13·6-s + 4.00e13i·7-s + 3.61e14i·8-s − 1.11e16·9-s + 2.48e17·11-s + 1.44e18i·12-s + 3.61e18i·13-s + 5.62e18·14-s − 4.50e19·16-s + 5.03e19i·17-s + 1.56e21i·18-s + 1.19e21·19-s + ⋯
L(s)  = 1  − 1.51i·2-s − 1.73i·3-s − 1.29·4-s − 2.62·6-s + 0.455i·7-s + 0.454i·8-s − 1.99·9-s + 1.62·11-s + 2.25i·12-s + 1.50i·13-s + 0.690·14-s − 0.610·16-s + 0.250i·17-s + 3.03i·18-s + 0.952·19-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

\(L(17)\) \(\approx\) \(2.415157962\)
\(L(\frac12)\) \(\approx\) \(2.415157962\)
\(L(\frac{35}{2})\) 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 \( 1 + 1.40e5iT - 8.58e9T^{2} \)
3 \( 1 + 1.29e8iT - 5.55e15T^{2} \)
7 \( 1 - 4.00e13iT - 7.73e27T^{2} \)
11 \( 1 - 2.48e17T + 2.32e34T^{2} \)
13 \( 1 - 3.61e18iT - 5.75e36T^{2} \)
17 \( 1 - 5.03e19iT - 4.02e40T^{2} \)
19 \( 1 - 1.19e21T + 1.58e42T^{2} \)
23 \( 1 + 2.35e22iT - 8.65e44T^{2} \)
29 \( 1 - 2.45e24T + 1.81e48T^{2} \)
31 \( 1 + 2.88e24T + 1.64e49T^{2} \)
37 \( 1 + 3.58e25iT - 5.63e51T^{2} \)
41 \( 1 - 6.61e25T + 1.66e53T^{2} \)
43 \( 1 - 1.70e27iT - 8.02e53T^{2} \)
47 \( 1 + 4.53e27iT - 1.51e55T^{2} \)
53 \( 1 - 1.11e28iT - 7.96e56T^{2} \)
59 \( 1 + 1.06e29T + 2.74e58T^{2} \)
61 \( 1 - 9.42e28T + 8.23e58T^{2} \)
67 \( 1 + 2.59e30iT - 1.82e60T^{2} \)
71 \( 1 + 4.27e30T + 1.23e61T^{2} \)
73 \( 1 - 2.23e30iT - 3.08e61T^{2} \)
79 \( 1 - 2.21e31T + 4.18e62T^{2} \)
83 \( 1 - 2.60e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.65e32T + 2.13e64T^{2} \)
97 \( 1 - 1.37e32iT - 3.65e65T^{2} \)
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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

−11.21568407473582885367181388452, −9.450191813029330284860941922260, −8.609721414026117835949132759045, −6.99094121951569230307699470709, −6.26574205437165358403274498088, −4.36342571457970988577778261644, −3.06008350091247274887920104310, −2.01948645920868270114599268954, −1.41680394325747864399078736595, −0.65331345698262208407916681585, 0.78174637452262207470667774196, 3.16429017535869065262543833483, 4.11768721007211009668056480222, 5.10165863492477440570301963055, 5.93356373256349564826946120196, 7.26013752275328143859935050216, 8.538489698675482882771897676435, 9.418536034778580244284897410632, 10.42824220602752804048147035337, 11.71702150711283318527861510447

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