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Properties

Label 1-925-925.187-r0-0-0
Degree $1$
Conductor $925$
Sign $0.937 - 0.348i$
Analytic cond. $4.29568$
Root an. cond. $4.29568$
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  + (−0.990 + 0.139i)2-s + (0.694 − 0.719i)3-s + (0.961 − 0.275i)4-s + (−0.587 + 0.809i)6-s + (0.642 + 0.766i)7-s + (−0.913 + 0.406i)8-s + (−0.0348 − 0.999i)9-s + (0.978 + 0.207i)11-s + (0.469 − 0.882i)12-s + (0.615 − 0.788i)13-s + (−0.743 − 0.669i)14-s + (0.848 − 0.529i)16-s + (0.961 + 0.275i)17-s + (0.173 + 0.984i)18-s + (0.898 + 0.438i)19-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)2-s + (0.694 − 0.719i)3-s + (0.961 − 0.275i)4-s + (−0.587 + 0.809i)6-s + (0.642 + 0.766i)7-s + (−0.913 + 0.406i)8-s + (−0.0348 − 0.999i)9-s + (0.978 + 0.207i)11-s + (0.469 − 0.882i)12-s + (0.615 − 0.788i)13-s + (−0.743 − 0.669i)14-s + (0.848 − 0.529i)16-s + (0.961 + 0.275i)17-s + (0.173 + 0.984i)18-s + (0.898 + 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(4.29568\)
Root analytic conductor: \(4.29568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (0:\ ),\ 0.937 - 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.502082883 - 0.2699329254i\)
\(L(\frac12)\) \(\approx\) \(1.502082883 - 0.2699329254i\)
\(L(1)\) \(\approx\) \(1.067225959 - 0.1334781642i\)
\(L(1)\) \(\approx\) \(1.067225959 - 0.1334781642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.990 + 0.139i)T \)
3 \( 1 + (0.694 - 0.719i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (0.961 + 0.275i)T \)
19 \( 1 + (0.898 + 0.438i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.615 - 0.788i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (-0.0697 + 0.997i)T \)
59 \( 1 + (0.529 + 0.848i)T \)
61 \( 1 + (0.999 + 0.0348i)T \)
67 \( 1 + (0.0697 + 0.997i)T \)
71 \( 1 + (-0.719 - 0.694i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.970 - 0.241i)T \)
83 \( 1 + (0.275 - 0.961i)T \)
89 \( 1 + (0.927 + 0.374i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−21.54089097002229594879731286757, −20.92364620797402824445959075738, −20.25524228063244342359496355413, −19.68288807172835484778862296917, −18.83245729974275908633432755697, −18.03669467573272735980183348132, −16.99642373128306073518915755638, −16.45463556921944346136014528882, −15.853795671609363898988895324214, −14.59539033830988526645034426456, −14.27584392103015502761292242378, −13.231034824839842279418400973632, −11.705330147608245791664001569306, −11.370942513109755238300770009038, −10.29514919900664027848323379221, −9.71264357949678200340460451430, −8.8536680173236633216704005553, −8.18030055557652665796575202391, −7.34674727961184177682480003198, −6.444086864853910191289898317477, −5.07776832353764776247862836056, −3.92230266517985931662277169490, −3.30409524470372074870151160006, −1.98590831892812742413889166972, −1.10692953373339159955917586375, 1.15505512684873916608135407255, 1.720331783779276494370272269132, 2.85049181000625577070465010548, 3.76797114841787584186406715166, 5.635694939889816559092826881698, 6.090733309556618375147426845879, 7.3959642490739278539225667955, 7.85753335144213097403808850237, 8.67042956464636155933189848019, 9.377498248119354433487364309029, 10.18201979147894645004606151346, 11.57094762422875029895039298727, 11.87018826030065742935912304461, 12.87281997853617217260506258566, 14.081556225271726015737906788793, 14.74586474334097224458455417148, 15.3810080190141331524085867196, 16.33161779936293664905813431006, 17.48859409113326412845429779988, 17.85641012773321334933279605675, 18.782308205949116612549825472097, 19.15226982584666136204087211930, 20.308063264259375178712874080713, 20.55058197854084240257770610219, 21.55439089438760101866893432416

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