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Properties

Label 1-35e2-1225.841-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.995 + 0.0921i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (−0.995 + 0.0896i)2-s + (−0.0448 + 0.998i)3-s + (0.983 − 0.178i)4-s + (−0.0448 − 0.998i)6-s + (−0.963 + 0.266i)8-s + (−0.995 − 0.0896i)9-s + (−0.995 + 0.0896i)11-s + (0.134 + 0.990i)12-s + (0.858 + 0.512i)13-s + (0.936 − 0.351i)16-s + (0.983 + 0.178i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.983 − 0.178i)22-s + (0.134 − 0.990i)23-s + (−0.222 − 0.974i)24-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (−0.0448 + 0.998i)3-s + (0.983 − 0.178i)4-s + (−0.0448 − 0.998i)6-s + (−0.963 + 0.266i)8-s + (−0.995 − 0.0896i)9-s + (−0.995 + 0.0896i)11-s + (0.134 + 0.990i)12-s + (0.858 + 0.512i)13-s + (0.936 − 0.351i)16-s + (0.983 + 0.178i)17-s + 18-s + (−0.809 − 0.587i)19-s + (0.983 − 0.178i)22-s + (0.134 − 0.990i)23-s + (−0.222 − 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.995 + 0.0921i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.995 + 0.0921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7425542250 + 0.03430228905i\)
\(L(\frac12)\) \(\approx\) \(0.7425542250 + 0.03430228905i\)
\(L(1)\) \(\approx\) \(0.6296142451 + 0.1595405901i\)
\(L(1)\) \(\approx\) \(0.6296142451 + 0.1595405901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (-0.0448 + 0.998i)T \)
11 \( 1 + (-0.995 + 0.0896i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.983 + 0.178i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.134 - 0.990i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.134 + 0.990i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.393 - 0.919i)T \)
53 \( 1 + (0.983 - 0.178i)T \)
59 \( 1 + (0.936 - 0.351i)T \)
61 \( 1 + (-0.691 - 0.722i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.858 - 0.512i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (0.858 - 0.512i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.939361846021288074506818764445, −20.14269035062312565746712600160, −19.40326477080681332977208498536, −18.69302158260923088215963181342, −18.14241516767331330266163956769, −17.597940947592903576186954947669, −16.60813498916177112593040375794, −16.03290613940744047714730878257, −15.04045669452803685029492888402, −14.157364227493806791850698911317, −13.071846680562834715192266559095, −12.561927394603960846203332922224, −11.68262623269942377516438852061, −10.814475386978337220375509656709, −10.27293175367464671597227928946, −9.05079385884250018401186467443, −8.336934722523667063864318231417, −7.67881000906872059226307982797, −7.02963529815693839058066784199, −5.93209564619603799288107587317, −5.44302703951933929824950079934, −3.53703836297608772255463366074, −2.78242313037605842555739251382, −1.73765849788189888930979367871, −0.91257705027795103815174920870, 0.52120358776497356768843894275, 2.0593116197668266031387302870, 2.93697889371753514455331937573, 3.95632706713495182200804069514, 5.0598671616822564100208490223, 5.941726662284761455597742188590, 6.742824217676121878077734548, 7.99939710292507914715972320274, 8.487018105698944162302787122892, 9.360816127537660348322676199754, 10.15116127144414561829970355247, 10.72495956406537981460542696812, 11.39902754807054559279568404889, 12.32437049452383220883763921254, 13.460513848547636600795762916481, 14.5520287932945147384033992566, 15.244286031423532201488748313533, 15.85616429038306761791395973404, 16.6449010359565931500337779240, 17.07755103579052420122731361710, 18.16117270196141277985889369625, 18.728422760629421641008394536465, 19.59300741786329123488946738179, 20.47864282017200124035106881957, 21.07982536301747867397619408219

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