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Properties

Label 2-5e2-5.3-c2-0-0
Degree $2$
Conductor $25$
Sign $0.437 - 0.899i$
Analytic cond. $0.681200$
Root an. cond. $0.825348$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)2-s + (1.22 + 1.22i)3-s + 1.00i·4-s − 2.99·6-s + (4.89 − 4.89i)7-s + (−6.12 − 6.12i)8-s − 6i·9-s − 3·11-s + (−1.22 + 1.22i)12-s + (7.34 + 7.34i)13-s + 11.9i·14-s + 10.9·16-s + (−13.4 + 13.4i)17-s + (7.34 + 7.34i)18-s − 5i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.612i)2-s + (0.408 + 0.408i)3-s + 0.250i·4-s − 0.499·6-s + (0.699 − 0.699i)7-s + (−0.765 − 0.765i)8-s − 0.666i·9-s − 0.272·11-s + (−0.102 + 0.102i)12-s + (0.565 + 0.565i)13-s + 0.857i·14-s + 0.687·16-s + (−0.792 + 0.792i)17-s + (0.408 + 0.408i)18-s − 0.263i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(0.681200\)
Root analytic conductor: \(0.825348\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.672853 + 0.420861i\)
\(L(\frac12)\) \(\approx\) \(0.672853 + 0.420861i\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T - 4iT^{2} \)
3 \( 1 + (-1.22 - 1.22i)T + 9iT^{2} \)
7 \( 1 + (-4.89 + 4.89i)T - 49iT^{2} \)
11 \( 1 + 3T + 121T^{2} \)
13 \( 1 + (-7.34 - 7.34i)T + 169iT^{2} \)
17 \( 1 + (13.4 - 13.4i)T - 289iT^{2} \)
19 \( 1 + 5iT - 361T^{2} \)
23 \( 1 + (17.1 + 17.1i)T + 529iT^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + 38T + 961T^{2} \)
37 \( 1 + (19.5 - 19.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 57T + 1.68e3T^{2} \)
43 \( 1 + (4.89 + 4.89i)T + 1.84e3iT^{2} \)
47 \( 1 + (7.34 - 7.34i)T - 2.20e3iT^{2} \)
53 \( 1 + (-31.8 - 31.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 90iT - 3.48e3T^{2} \)
61 \( 1 + 28T + 3.72e3T^{2} \)
67 \( 1 + (-47.7 + 47.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 42T + 5.04e3T^{2} \)
73 \( 1 + (-13.4 - 13.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 80iT - 6.24e3T^{2} \)
83 \( 1 + (-111. - 111. i)T + 6.88e3iT^{2} \)
89 \( 1 - 15iT - 7.92e3T^{2} \)
97 \( 1 + (-53.8 + 53.8i)T - 9.40e3iT^{2} \)
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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

−17.56119575775530289336883258965, −16.42661081484645235263474578445, −15.35474335332985474713232506729, −14.17692961093489109108404805037, −12.57910233808340025768232932523, −10.87036483367697919382493644174, −9.231979411563966906372676961873, −8.181050810439558547923718676808, −6.66659017825642368793563998673, −3.96617731422108493669297013406, 2.18276126975121981967295505990, 5.50779105914296339937998810838, 7.902577928817560403427339720800, 9.075493128417653782841684094869, 10.65256669380755182035270122951, 11.72130267965928507532906959660, 13.38291394650940071904926706610, 14.61987166921431862955652160781, 15.87327121129670108706619144263, 17.80124358464963284926715165367

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