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Properties

Label 2-425-5.4-c3-0-48
Degree $2$
Conductor $425$
Sign $-0.894 + 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·2-s − 5i·3-s − 4-s − 15·6-s + 22i·7-s − 21i·8-s + 2·9-s + 60·11-s + 5i·12-s − 31i·13-s + 66·14-s − 71·16-s − 17i·17-s − 6i·18-s + 61·19-s + ⋯
L(s)  = 1  − 1.06i·2-s − 0.962i·3-s − 0.125·4-s − 1.02·6-s + 1.18i·7-s − 0.928i·8-s + 0.0740·9-s + 1.64·11-s + 0.120i·12-s − 0.661i·13-s + 1.25·14-s − 1.10·16-s − 0.242i·17-s − 0.0785i·18-s + 0.736·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.396821308\)
\(L(\frac12)\) \(\approx\) \(2.396821308\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + 17iT \)
good2 \( 1 + 3iT - 8T^{2} \)
3 \( 1 + 5iT - 27T^{2} \)
7 \( 1 - 22iT - 343T^{2} \)
11 \( 1 - 60T + 1.33e3T^{2} \)
13 \( 1 + 31iT - 2.19e3T^{2} \)
19 \( 1 - 61T + 6.85e3T^{2} \)
23 \( 1 + 78iT - 1.21e4T^{2} \)
29 \( 1 + 69T + 2.43e4T^{2} \)
31 \( 1 + 31T + 2.97e4T^{2} \)
37 \( 1 + 56iT - 5.06e4T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 + 538iT - 7.95e4T^{2} \)
47 \( 1 - 465iT - 1.03e5T^{2} \)
53 \( 1 - 723iT - 1.48e5T^{2} \)
59 \( 1 - 753T + 2.05e5T^{2} \)
61 \( 1 - 35T + 2.26e5T^{2} \)
67 \( 1 - 322iT - 3.00e5T^{2} \)
71 \( 1 + 99T + 3.57e5T^{2} \)
73 \( 1 + 1.12e3iT - 3.89e5T^{2} \)
79 \( 1 + 488T + 4.93e5T^{2} \)
83 \( 1 + 852iT - 5.71e5T^{2} \)
89 \( 1 + 1.21e3T + 7.04e5T^{2} \)
97 \( 1 - 601iT - 9.12e5T^{2} \)
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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

−10.54754195189136177944178651436, −9.482007864805653979468456546494, −8.816212109114210513782772180720, −7.48372947879836718651895072872, −6.64709698384608832953484020100, −5.73771771915643621506658024951, −4.10184364383640938665762479633, −2.86518839373075151352006329301, −1.85887681046350446068606899067, −0.874448559364318066784218900091, 1.45504075577426056810832806750, 3.63696255267406558449950485915, 4.33241890101119398733755401396, 5.44588159960337120073855391967, 6.74294886470314225379957839295, 7.11505561263418080937064605666, 8.323879995802266934735424163363, 9.391856245054443361365465584282, 9.994503303942729494075999201032, 11.21219505456552640037108400351

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