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Properties

Label 2-725-5.4-c1-0-38
Degree $2$
Conductor $725$
Sign $-0.447 - 0.894i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.17i·2-s − 1.70i·3-s − 2.70·4-s − 3.70·6-s − 3.70i·7-s + 1.53i·8-s + 0.0783·9-s − 0.630·11-s + 4.63i·12-s − 4.34i·13-s − 8.04·14-s − 2.07·16-s + 1.55i·17-s − 0.170i·18-s + 5.70·19-s + ⋯
L(s)  = 1  − 1.53i·2-s − 0.986i·3-s − 1.35·4-s − 1.51·6-s − 1.40i·7-s + 0.544i·8-s + 0.0261·9-s − 0.190·11-s + 1.33i·12-s − 1.20i·13-s − 2.15·14-s − 0.519·16-s + 0.376i·17-s − 0.0400i·18-s + 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726635 + 1.17572i\)
\(L(\frac12)\) \(\approx\) \(0.726635 + 1.17572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + 2.17iT - 2T^{2} \)
3 \( 1 + 1.70iT - 3T^{2} \)
7 \( 1 + 3.70iT - 7T^{2} \)
11 \( 1 + 0.630T + 11T^{2} \)
13 \( 1 + 4.34iT - 13T^{2} \)
17 \( 1 - 1.55iT - 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 - 6.63iT - 23T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + 2.29iT - 47T^{2} \)
53 \( 1 - 0.921iT - 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 3.12iT - 83T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−10.05452179481816535205357440352, −9.397801708910328692825188253695, −7.80798568270954976450337556965, −7.55326983954745052367097198208, −6.37086039702068959617762122837, −4.98645972511847898809281148467, −3.80260871876234444389303564545, −3.00187261787455521960232466110, −1.54809453757159225276825775079, −0.77885041424607307335809384620, 2.44240073077670319030606803441, 4.02562367716635725610909953401, 5.00161255056404418394569425802, 5.54872122696330979786884630236, 6.57444527007001598289700954134, 7.38374999331820680058525237904, 8.509159514775211764793564870564, 9.132032771540107266395525392808, 9.632901761664393900380847110488, 10.85898172398336842862309935308

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