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Properties

Label 2-425-5.4-c3-0-4
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 − 7i·3-s − 4-s − 21·6-s + 22i·7-s − 21i·8-s − 22·9-s − 64·11-s + 7i·12-s + 73i·13-s + 66·14-s − 71·16-s + 17i·17-s + 66i·18-s + 49·19-s + ⋯
L(s)  = 1  − 1.06i·2-s − 1.34i·3-s − 0.125·4-s − 1.42·6-s + 1.18i·7-s − 0.928i·8-s − 0.814·9-s − 1.75·11-s + 0.168i·12-s + 1.55i·13-s + 1.25·14-s − 1.10·16-s + 0.242i·17-s + 0.864i·18-s + 0.591·19-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

\(L(2)\) \(\approx\) \(0.5121180686\)
\(L(\frac12)\) \(\approx\) \(0.5121180686\)
\(L(\frac{5}{2})\) 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 \( 1 + 7iT - 27T^{2} \)
7 \( 1 - 22iT - 343T^{2} \)
11 \( 1 + 64T + 1.33e3T^{2} \)
13 \( 1 - 73iT - 2.19e3T^{2} \)
19 \( 1 - 49T + 6.85e3T^{2} \)
23 \( 1 - 110iT - 1.21e4T^{2} \)
29 \( 1 + 155T + 2.43e4T^{2} \)
31 \( 1 + 197T + 2.97e4T^{2} \)
37 \( 1 - 372iT - 5.06e4T^{2} \)
41 \( 1 + 262T + 6.89e4T^{2} \)
43 \( 1 - 258iT - 7.95e4T^{2} \)
47 \( 1 - 13iT - 1.03e5T^{2} \)
53 \( 1 + 653iT - 1.48e5T^{2} \)
59 \( 1 - 333T + 2.05e5T^{2} \)
61 \( 1 + 355T + 2.26e5T^{2} \)
67 \( 1 + 814iT - 3.00e5T^{2} \)
71 \( 1 - 47T + 3.57e5T^{2} \)
73 \( 1 + 437iT - 3.89e5T^{2} \)
79 \( 1 - 384T + 4.93e5T^{2} \)
83 \( 1 + 736iT - 5.71e5T^{2} \)
89 \( 1 + 511T + 7.04e5T^{2} \)
97 \( 1 + 537iT - 9.12e5T^{2} \)
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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

−11.26686393124493063682663692122, −10.00895928951662892675725726915, −9.143450675301090011209132432820, −7.997802593812253972865844988155, −7.15870408777292950918597690960, −6.20843508144894216410643501484, −5.08393073675369777514031528193, −3.29657254758631081852671379596, −2.20060047896779106553108096465, −1.64490016320848755474632482515, 0.15215643163665453418015397651, 2.75934179290541151310672882166, 3.97028971865974904010959001297, 5.23277594110335367202037789529, 5.54211505773935280557794621561, 7.24380984612948277756617058076, 7.68959009995579330660960144869, 8.721083287934512315851003840689, 9.969677708016068659007001447033, 10.67648762926406493718688872876

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