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Properties

Label 1-35e2-1225.158-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.486 - 0.873i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (0.850 − 0.525i)2-s + (0.999 + 0.0149i)3-s + (0.447 − 0.894i)4-s + (0.858 − 0.512i)6-s + (−0.0896 − 0.995i)8-s + (0.999 + 0.0299i)9-s + (−0.999 + 0.0299i)11-s + (0.460 − 0.887i)12-s + (−0.178 + 0.983i)13-s + (−0.599 − 0.800i)16-s + (0.0598 − 0.998i)17-s + (0.866 − 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.834 + 0.550i)22-s + (0.538 − 0.842i)23-s + (−0.0747 − 0.997i)24-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (0.999 + 0.0149i)3-s + (0.447 − 0.894i)4-s + (0.858 − 0.512i)6-s + (−0.0896 − 0.995i)8-s + (0.999 + 0.0299i)9-s + (−0.999 + 0.0299i)11-s + (0.460 − 0.887i)12-s + (−0.178 + 0.983i)13-s + (−0.599 − 0.800i)16-s + (0.0598 − 0.998i)17-s + (0.866 − 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.834 + 0.550i)22-s + (0.538 − 0.842i)23-s + (−0.0747 − 0.997i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.627890012 - 4.469641139i\)
\(L(\frac12)\) \(\approx\) \(2.627890012 - 4.469641139i\)
\(L(1)\) \(\approx\) \(2.073587524 - 1.132163962i\)
\(L(1)\) \(\approx\) \(2.073587524 - 1.132163962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.850 - 0.525i)T \)
3 \( 1 + (0.999 + 0.0149i)T \)
11 \( 1 + (-0.999 + 0.0299i)T \)
13 \( 1 + (-0.178 + 0.983i)T \)
17 \( 1 + (0.0598 - 0.998i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.538 - 0.842i)T \)
29 \( 1 + (-0.936 + 0.351i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.460 - 0.887i)T \)
41 \( 1 + (0.753 - 0.657i)T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (-0.379 + 0.925i)T \)
53 \( 1 + (-0.894 - 0.447i)T \)
59 \( 1 + (-0.992 + 0.119i)T \)
61 \( 1 + (0.712 - 0.701i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (0.762 - 0.646i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.990 + 0.134i)T \)
89 \( 1 + (0.337 - 0.941i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−21.24711625830387240224148443843, −20.52639585860223421836087781636, −19.928105272743856809485299864863, −19.02488756601332773318192248524, −18.02301836447002272642112694215, −17.3481010986110582899666122518, −16.22191098732370762713448344485, −15.524542785267979946116849552466, −15.07383633912016897658400596449, −14.29356538398360348182171389670, −13.287893944121489291932414405947, −13.10457948011006668675911187043, −12.18499856171696601670079685840, −11.06511881978997905292661886759, −10.1727198559195618880616752389, −9.20009284915128068599376861828, −8.048079221759074198773322854302, −7.83700466414530195193125449094, −6.90853124923606344033339181796, −5.76210891808162206909605422371, −5.04450821849280201029814006022, −4.03515686187987137397489261772, −3.1066930065232454189119347793, −2.62326359448248436364718835694, −1.31524058558233457826413923483, 0.60696551971251336814412375072, 1.85554136344747636089427760719, 2.60803503584816016197087578164, 3.312332072622307481248053758206, 4.37084574978772731514055850885, 4.98313500923379414694932242610, 6.09714220801035310274128117453, 7.215683332620996264440938048746, 7.72776548064796623019850405492, 9.17580910163512346073036018854, 9.53656109127516875824781298255, 10.56779945141311980846612996958, 11.33255254031442114847021850705, 12.34491752780370403620273114986, 12.97481553516795650759324333744, 13.8360693754390314743352510981, 14.24888788082038635984264577185, 15.073406541903087321589557803375, 15.9174609749692576996271247147, 16.394231992834213722196498045816, 18.00519936299834683939658455060, 18.75824741170206243837399793610, 19.18701125961883658242907843441, 20.205757548686310352109910344753, 20.82325852671764069984238360434

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