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Properties

Label 1-65e2-4225.673-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.0978 + 0.995i$
Analytic cond. $454.039$
Root an. cond. $454.039$
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  + (0.561 − 0.827i)2-s + (−0.857 + 0.513i)3-s + (−0.369 − 0.929i)4-s + (−0.0563 + 0.998i)6-s + (−0.316 + 0.948i)7-s + (−0.976 − 0.215i)8-s + (0.471 − 0.881i)9-s + (0.471 + 0.881i)11-s + (0.794 + 0.607i)12-s + (0.607 + 0.794i)14-s + (−0.726 + 0.686i)16-s + (0.301 + 0.953i)17-s + (−0.464 − 0.885i)18-s + (0.913 + 0.406i)19-s + (−0.215 − 0.976i)21-s + (0.994 + 0.104i)22-s + ⋯
L(s)  = 1  + (0.561 − 0.827i)2-s + (−0.857 + 0.513i)3-s + (−0.369 − 0.929i)4-s + (−0.0563 + 0.998i)6-s + (−0.316 + 0.948i)7-s + (−0.976 − 0.215i)8-s + (0.471 − 0.881i)9-s + (0.471 + 0.881i)11-s + (0.794 + 0.607i)12-s + (0.607 + 0.794i)14-s + (−0.726 + 0.686i)16-s + (0.301 + 0.953i)17-s + (−0.464 − 0.885i)18-s + (0.913 + 0.406i)19-s + (−0.215 − 0.976i)21-s + (0.994 + 0.104i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0978 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0978 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.0978 + 0.995i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.0978 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443942244 + 1.308906507i\)
\(L(\frac12)\) \(\approx\) \(1.443942244 + 1.308906507i\)
\(L(1)\) \(\approx\) \(1.041608813 + 0.01824770348i\)
\(L(1)\) \(\approx\) \(1.041608813 + 0.01824770348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.561 - 0.827i)T \)
3 \( 1 + (-0.857 + 0.513i)T \)
7 \( 1 + (-0.316 + 0.948i)T \)
11 \( 1 + (0.471 + 0.881i)T \)
17 \( 1 + (0.301 + 0.953i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.743 + 0.669i)T \)
29 \( 1 + (0.789 - 0.613i)T \)
31 \( 1 + (-0.836 - 0.548i)T \)
37 \( 1 + (-0.988 + 0.152i)T \)
41 \( 1 + (-0.0884 + 0.996i)T \)
43 \( 1 + (0.160 + 0.987i)T \)
47 \( 1 + (0.896 - 0.443i)T \)
53 \( 1 + (0.0965 + 0.995i)T \)
59 \( 1 + (0.991 + 0.128i)T \)
61 \( 1 + (0.870 + 0.493i)T \)
67 \( 1 + (0.929 + 0.369i)T \)
71 \( 1 + (0.999 - 0.0161i)T \)
73 \( 1 + (0.849 + 0.527i)T \)
79 \( 1 + (0.443 + 0.896i)T \)
83 \( 1 + (0.873 - 0.485i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.650 + 0.759i)T \)
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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

−17.79054316737964629311364308519, −17.31995141075449945961502634484, −16.50026177298112738382957648558, −16.18490960922771677617432441541, −15.68796436314868874980939499438, −14.31778119015007335090578417695, −13.93352733816932010380376556753, −13.54681779599413323476555210498, −12.55079490101850178766243404249, −12.143452343517497424229403141, −11.34171337126296128041448124441, −10.677452782287243298413269193162, −9.790271958674218829668373379832, −8.86501353861281973868841525488, −8.085194034780667867172545647012, −7.17517077644748228372887199084, −6.95070621560760705721388935229, −6.19885354560843210637029782709, −5.34091372875892204599230874690, −4.92682366850701215406784548934, −3.84620952480235280973517727966, −3.33859870993178326446553115933, −2.16849750638952078539959332237, −0.78008168465741035696293782472, −0.408008349125158065214397087623, 0.89265901117256506594891626141, 1.696962270387223182584148920647, 2.543515149028550742976276392305, 3.64612100186786908454504437074, 3.99833724815387153609686317790, 5.01164321376117600946331335556, 5.55241906431049154458884204075, 6.153638733585544837607407971810, 6.85065042605776855476996500777, 8.06808050299051053450174736726, 9.094056468629710225076038665060, 9.76562663419484514781537107313, 10.01870528218810781399521659888, 10.97764927887273421312638820809, 11.702003597250794486818428091206, 12.16333855727201604621226498671, 12.5801605539410076084208879242, 13.43007353632175317079532045211, 14.405520950514315831911437204870, 14.9911830383747908631535972682, 15.55496732205561324921429280970, 16.16792537480416954301653252110, 17.13965128474512399007323963319, 17.8267110007880178421008160069, 18.341657748329046120463208400456

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