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Properties

Label 1-425-425.131-r1-0-0
Degree $1$
Conductor $425$
Sign $-0.0676 + 0.997i$
Analytic cond. $45.6725$
Root an. cond. $45.6725$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.760 − 0.649i)3-s + (−0.951 − 0.309i)4-s + (−0.760 + 0.649i)6-s + (0.923 − 0.382i)7-s + (−0.453 + 0.891i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.522 + 0.852i)12-s + (0.587 − 0.809i)13-s + (−0.233 − 0.972i)14-s + (0.809 + 0.587i)16-s + 18-s + (−0.891 − 0.453i)19-s + (−0.951 − 0.309i)21-s + (0.649 − 0.760i)22-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.760 − 0.649i)3-s + (−0.951 − 0.309i)4-s + (−0.760 + 0.649i)6-s + (0.923 − 0.382i)7-s + (−0.453 + 0.891i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.522 + 0.852i)12-s + (0.587 − 0.809i)13-s + (−0.233 − 0.972i)14-s + (0.809 + 0.587i)16-s + 18-s + (−0.891 − 0.453i)19-s + (−0.951 − 0.309i)21-s + (0.649 − 0.760i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0676 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0676 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.0676 + 0.997i$
Analytic conductor: \(45.6725\)
Root analytic conductor: \(45.6725\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 425,\ (1:\ ),\ -0.0676 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2966253452 - 0.3174217176i\)
\(L(\frac12)\) \(\approx\) \(-0.2966253452 - 0.3174217176i\)
\(L(1)\) \(\approx\) \(0.5462392375 - 0.5548908375i\)
\(L(1)\) \(\approx\) \(0.5462392375 - 0.5548908375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.156 - 0.987i)T \)
3 \( 1 + (-0.760 - 0.649i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.852 - 0.522i)T \)
29 \( 1 + (-0.649 + 0.760i)T \)
31 \( 1 + (-0.996 - 0.0784i)T \)
37 \( 1 + (-0.852 + 0.522i)T \)
41 \( 1 + (-0.972 - 0.233i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (-0.156 - 0.987i)T \)
61 \( 1 + (0.522 - 0.852i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.760 - 0.649i)T \)
73 \( 1 + (-0.972 + 0.233i)T \)
79 \( 1 + (-0.996 + 0.0784i)T \)
83 \( 1 + (0.891 + 0.453i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (-0.649 + 0.760i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.39692873755762563822389764174, −23.780290999276047860394170547981, −23.037684098714670974709448998153, −21.965139300081003186275416602210, −21.56840154232944770393884972936, −20.67866448290754571912366005421, −19.083909448248455055219963780213, −18.22492123554605791094688524598, −17.41644424336734157897028356545, −16.681863190689424180330049999783, −16.01240593490397034364769295600, −14.96190462056636361460524165884, −14.43484161966760175788693774486, −13.35479256177443230684514057966, −12.00995026238815158650939293078, −11.45592016100504993001132971626, −10.23595239011319237830156054738, −9.06441918262979105231159389481, −8.49834229090092631599673492319, −7.146314809425625599640773523982, −6.0944809011127549116054561720, −5.52060661721271408436201165850, −4.308336466389172503871547448283, −3.753899277860106572037192938480, −1.51905424782094419390295640753, 0.13008876013535741271636117219, 1.3527852948767726881335202592, 2.07188175952835578960418001869, 3.72642890693434666230339090701, 4.71849011569620183888243102265, 5.610969048700916205087139606815, 6.79820262308171043693604926288, 7.989828067625075996608196140963, 8.9058923529238221811117302542, 10.38313308314116931251764727586, 10.8627218900523078417498819521, 11.79983407420780088563168583231, 12.49471979533881679159037868898, 13.39671327469718495379208004469, 14.237647739241914037254560025412, 15.20940362983215238240777662994, 16.81963239545676412502434678946, 17.469721960963727847880937007732, 18.17335031225478563862451345327, 18.92379824319862355943802602713, 20.09619238622989931269181288239, 20.49745612132822522819529147133, 21.8573318400334564123778997671, 22.29339930192062129730976074992, 23.38325395371194597689372530190

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