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Properties

Label 2-5e2-5.4-c33-0-24
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  + 7.64e4i·2-s − 7.43e7i·3-s + 2.75e9·4-s + 5.67e12·6-s + 1.36e14i·7-s + 8.66e14i·8-s + 3.57e13·9-s + 3.84e16·11-s − 2.04e17i·12-s − 2.57e18i·13-s − 1.04e19·14-s − 4.25e19·16-s + 1.44e20i·17-s + 2.73e18i·18-s + 2.79e20·19-s + ⋯
L(s)  = 1  + 0.824i·2-s − 0.996i·3-s + 0.320·4-s + 0.821·6-s + 1.55i·7-s + 1.08i·8-s + 0.00642·9-s + 0.252·11-s − 0.319i·12-s − 1.07i·13-s − 1.28·14-s − 0.577·16-s + 0.720i·17-s + 0.00530i·18-s + 0.222·19-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

\(L(17)\) \(\approx\) \(3.182370887\)
\(L(\frac12)\) \(\approx\) \(3.182370887\)
\(L(\frac{35}{2})\) 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 \( 1 - 7.64e4iT - 8.58e9T^{2} \)
3 \( 1 + 7.43e7iT - 5.55e15T^{2} \)
7 \( 1 - 1.36e14iT - 7.73e27T^{2} \)
11 \( 1 - 3.84e16T + 2.32e34T^{2} \)
13 \( 1 + 2.57e18iT - 5.75e36T^{2} \)
17 \( 1 - 1.44e20iT - 4.02e40T^{2} \)
19 \( 1 - 2.79e20T + 1.58e42T^{2} \)
23 \( 1 + 3.71e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.90e24T + 1.81e48T^{2} \)
31 \( 1 - 5.88e24T + 1.64e49T^{2} \)
37 \( 1 - 1.05e26iT - 5.63e51T^{2} \)
41 \( 1 + 2.64e25T + 1.66e53T^{2} \)
43 \( 1 + 1.20e27iT - 8.02e53T^{2} \)
47 \( 1 + 3.53e27iT - 1.51e55T^{2} \)
53 \( 1 - 3.25e28iT - 7.96e56T^{2} \)
59 \( 1 + 3.41e28T + 2.74e58T^{2} \)
61 \( 1 + 6.01e28T + 8.23e58T^{2} \)
67 \( 1 + 2.17e30iT - 1.82e60T^{2} \)
71 \( 1 - 9.20e29T + 1.23e61T^{2} \)
73 \( 1 - 4.72e30iT - 3.08e61T^{2} \)
79 \( 1 - 2.14e31T + 4.18e62T^{2} \)
83 \( 1 - 6.60e31iT - 2.13e63T^{2} \)
89 \( 1 - 6.08e31T + 2.13e64T^{2} \)
97 \( 1 - 6.12e30iT - 3.65e65T^{2} \)
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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

−11.95842458293673238218594657149, −10.35517050379939281349577884907, −8.551325465939482685609597598773, −8.018547793855302455882897811375, −6.63689536431008267932843371125, −6.09527098604472040248585648653, −4.96101649293083156194838971531, −2.84264474049876799803766404528, −2.11302401329779819948821669578, −0.904082207925808231484590523876, 0.71428932532107507233888450870, 1.54662376340999003432521017151, 3.04279117838924016502408632771, 3.98132036736082987192517466784, 4.63690811903605546975774135883, 6.58110640579662807537191488007, 7.47505134527133313145663857651, 9.420774644619564592822036499981, 10.07190519106831940359062980562, 10.97455765625558885020859228981

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