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Properties

Label 2-725-5.4-c1-0-32
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  + 2.41i·2-s − 2i·3-s − 3.82·4-s + 4.82·6-s − 0.828i·7-s − 4.41i·8-s − 9-s − 4.82·11-s + 7.65i·12-s − 2i·13-s + 1.99·14-s + 2.99·16-s + 2.82i·17-s − 2.41i·18-s − 0.828·19-s + ⋯
L(s)  = 1  + 1.70i·2-s − 1.15i·3-s − 1.91·4-s + 1.97·6-s − 0.313i·7-s − 1.56i·8-s − 0.333·9-s − 1.45·11-s + 2.21i·12-s − 0.554i·13-s + 0.534·14-s + 0.749·16-s + 0.685i·17-s − 0.569i·18-s − 0.190·19-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

\(L(1)\) \(\approx\) \(0.522416 - 0.322871i\)
\(L(\frac12)\) \(\approx\) \(0.522416 - 0.322871i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 2T^{2} \)
3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 + 8.82iT - 23T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 4.48iT - 97T^{2} \)
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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

−10.17569668213723855639152987294, −8.778722661615424846564226863038, −8.197256247927807152361095799086, −7.41880192803429992872452724615, −6.98877572090141888045934799083, −5.95619437309432876618056402143, −5.31323269119523513982193402981, −4.09903684341191952450336855298, −2.30385375557248273607590224548, −0.30620756861729516670729676068, 1.82655675637076912049312911936, 3.03205754067048600552143009051, 3.78146777540943074370420425455, 4.87357471908976139441332310904, 5.41054621581820714585244244112, 7.25187404212121187614104464163, 8.477990042359555698814321120745, 9.429539126246900236565855659121, 9.791916847456475860649503817538, 10.58398900159999843837438598884

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