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Properties

Label 2-5e2-5.4-c33-0-36
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  + 5.62e3i·2-s + 1.24e8i·3-s + 8.55e9·4-s − 7.01e11·6-s + 7.01e13i·7-s + 9.65e13i·8-s − 9.96e15·9-s + 9.39e16·11-s + 1.06e18i·12-s − 9.95e17i·13-s − 3.94e17·14-s + 7.29e19·16-s − 2.59e20i·17-s − 5.60e19i·18-s + 1.31e21·19-s + ⋯
L(s)  = 1  + 0.0607i·2-s + 1.67i·3-s + 0.996·4-s − 0.101·6-s + 0.797i·7-s + 0.121i·8-s − 1.79·9-s + 0.616·11-s + 1.66i·12-s − 0.414i·13-s − 0.0484·14-s + 0.988·16-s − 1.29i·17-s − 0.108i·18-s + 1.04·19-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

\(L(17)\) \(\approx\) \(3.501416361\)
\(L(\frac12)\) \(\approx\) \(3.501416361\)
\(L(\frac{35}{2})\) 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 \( 1 - 5.62e3iT - 8.58e9T^{2} \)
3 \( 1 - 1.24e8iT - 5.55e15T^{2} \)
7 \( 1 - 7.01e13iT - 7.73e27T^{2} \)
11 \( 1 - 9.39e16T + 2.32e34T^{2} \)
13 \( 1 + 9.95e17iT - 5.75e36T^{2} \)
17 \( 1 + 2.59e20iT - 4.02e40T^{2} \)
19 \( 1 - 1.31e21T + 1.58e42T^{2} \)
23 \( 1 + 5.44e22iT - 8.65e44T^{2} \)
29 \( 1 - 9.77e23T + 1.81e48T^{2} \)
31 \( 1 - 6.82e24T + 1.64e49T^{2} \)
37 \( 1 + 7.50e25iT - 5.63e51T^{2} \)
41 \( 1 + 2.94e25T + 1.66e53T^{2} \)
43 \( 1 - 1.00e27iT - 8.02e53T^{2} \)
47 \( 1 - 1.91e27iT - 1.51e55T^{2} \)
53 \( 1 + 4.33e28iT - 7.96e56T^{2} \)
59 \( 1 - 1.21e29T + 2.74e58T^{2} \)
61 \( 1 - 4.03e29T + 8.23e58T^{2} \)
67 \( 1 + 1.33e30iT - 1.82e60T^{2} \)
71 \( 1 + 4.10e30T + 1.23e61T^{2} \)
73 \( 1 + 8.64e30iT - 3.08e61T^{2} \)
79 \( 1 - 1.76e31T + 4.18e62T^{2} \)
83 \( 1 - 6.43e30iT - 2.13e63T^{2} \)
89 \( 1 + 1.32e32T + 2.13e64T^{2} \)
97 \( 1 - 2.90e32iT - 3.65e65T^{2} \)
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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

−11.37457624200705690036965624903, −10.29453979336114417207640838854, −9.413023116316692837354808948990, −8.265871378837117325094884173338, −6.62965439695761274061499760034, −5.49105335356837403521395675583, −4.54118412049803865563068889992, −3.14983539169114755508582808051, −2.51639441082317849012827504900, −0.70880989843301327766429578592, 1.13026141859663520813906629568, 1.29799044623706327913262025024, 2.51975809631545684432156559925, 3.72320698016225101047829869458, 5.76448490846538642097008050016, 6.72131655376044296631880128763, 7.33318537824201402543098996983, 8.312282622101318622739509620271, 10.12984277748803879123659890003, 11.54417987967536605350454588597

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