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Properties

Label 2-5e2-5.3-c32-0-6
Degree $2$
Conductor $25$
Sign $0.850 - 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.95e4 − 3.95e4i)2-s + (−4.69e7 − 4.69e7i)3-s + 1.16e9i·4-s − 3.71e12·6-s + (−2.85e13 + 2.85e13i)7-s + (2.16e14 + 2.16e14i)8-s + 2.55e15i·9-s + 6.29e16·11-s + (5.48e16 − 5.48e16i)12-s + (−6.68e17 − 6.68e17i)13-s + 2.25e18i·14-s + 1.20e19·16-s + (−9.86e17 + 9.86e17i)17-s + (1.01e20 + 1.01e20i)18-s − 3.77e20i·19-s + ⋯
L(s)  = 1  + (0.603 − 0.603i)2-s + (−1.09 − 1.09i)3-s + 0.272i·4-s − 1.31·6-s + (−0.858 + 0.858i)7-s + (0.767 + 0.767i)8-s + 1.37i·9-s + 1.36·11-s + (0.296 − 0.296i)12-s + (−1.00 − 1.00i)13-s + 1.03i·14-s + 0.653·16-s + (−0.0202 + 0.0202i)17-s + (0.832 + 0.832i)18-s − 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.9042415567\)
\(L(\frac12)\) \(\approx\) \(0.9042415567\)
\(L(17)\) 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 \( 1 + (-3.95e4 + 3.95e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (4.69e7 + 4.69e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (2.85e13 - 2.85e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 6.29e16T + 2.11e33T^{2} \)
13 \( 1 + (6.68e17 + 6.68e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (9.86e17 - 9.86e17i)T - 2.36e39iT^{2} \)
19 \( 1 + 3.77e20iT - 8.31e40T^{2} \)
23 \( 1 + (5.92e21 + 5.92e21i)T + 3.76e43iT^{2} \)
29 \( 1 - 2.77e19iT - 6.26e46T^{2} \)
31 \( 1 + 3.93e23T + 5.29e47T^{2} \)
37 \( 1 + (-4.19e24 + 4.19e24i)T - 1.52e50iT^{2} \)
41 \( 1 - 8.53e25T + 4.06e51T^{2} \)
43 \( 1 + (-8.89e25 - 8.89e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (3.07e26 - 3.07e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (4.54e26 + 4.54e26i)T + 1.50e55iT^{2} \)
59 \( 1 - 1.68e28iT - 4.64e56T^{2} \)
61 \( 1 + 1.17e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.91e29 + 1.91e29i)T - 2.71e58iT^{2} \)
71 \( 1 + 1.41e29T + 1.73e59T^{2} \)
73 \( 1 + (5.53e29 + 5.53e29i)T + 4.22e59iT^{2} \)
79 \( 1 + 1.90e30iT - 5.29e60T^{2} \)
83 \( 1 + (2.78e30 + 2.78e30i)T + 2.57e61iT^{2} \)
89 \( 1 - 2.46e30iT - 2.40e62T^{2} \)
97 \( 1 + (4.72e31 - 4.72e31i)T - 3.77e63iT^{2} \)
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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

−11.94897320453717348483757582732, −10.91820987098162687590704845003, −9.261415870978548055941123094901, −7.69799401377048945073274600411, −6.59617530859423558269677338870, −5.69438916325505969002995874186, −4.46106201151734049583678161717, −2.95764259071864145098460155221, −2.10945736090410759023423965500, −0.77929500598027450893595965909, 0.21671457620440373438249615853, 1.47531786133233563009934125463, 3.97050387968858567150575434942, 4.09549132688700900876118691186, 5.47473159942193916758195358677, 6.31471609103563471312368298582, 7.17880339190226032478437134718, 9.637620127943638301442533917893, 9.929389353742410660165802965065, 11.24412944433187080801872526495

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