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Properties

Label 1-35e2-1225.659-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.939 - 0.341i$
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  + (−0.936 − 0.351i)2-s + (−0.983 + 0.178i)3-s + (0.753 + 0.657i)4-s + (0.983 + 0.178i)6-s + (−0.473 − 0.880i)8-s + (0.936 − 0.351i)9-s + (0.936 + 0.351i)11-s + (−0.858 − 0.512i)12-s + (0.550 + 0.834i)13-s + (0.134 + 0.990i)16-s + (−0.753 + 0.657i)17-s − 18-s + (−0.809 − 0.587i)19-s + (−0.753 − 0.657i)22-s + (−0.858 + 0.512i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (−0.936 − 0.351i)2-s + (−0.983 + 0.178i)3-s + (0.753 + 0.657i)4-s + (0.983 + 0.178i)6-s + (−0.473 − 0.880i)8-s + (0.936 − 0.351i)9-s + (0.936 + 0.351i)11-s + (−0.858 − 0.512i)12-s + (0.550 + 0.834i)13-s + (0.134 + 0.990i)16-s + (−0.753 + 0.657i)17-s − 18-s + (−0.809 − 0.587i)19-s + (−0.753 − 0.657i)22-s + (−0.858 + 0.512i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.939 - 0.341i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01277109671 - 0.07248888070i\)
\(L(\frac12)\) \(\approx\) \(0.01277109671 - 0.07248888070i\)
\(L(1)\) \(\approx\) \(0.4519123447 + 0.01555464137i\)
\(L(1)\) \(\approx\) \(0.4519123447 + 0.01555464137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.936 - 0.351i)T \)
3 \( 1 + (-0.983 + 0.178i)T \)
11 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (0.550 + 0.834i)T \)
17 \( 1 + (-0.753 + 0.657i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.858 + 0.512i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.858 - 0.512i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.0448 - 0.998i)T \)
53 \( 1 + (-0.753 - 0.657i)T \)
59 \( 1 + (0.134 + 0.990i)T \)
61 \( 1 + (-0.995 + 0.0896i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (0.550 - 0.834i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.0448 + 0.998i)T \)
89 \( 1 + (-0.550 + 0.834i)T \)
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

−21.51276933728026368894728304551, −20.450389899469973728756764002378, −19.87752320935290011695106678385, −18.78789309000249348999244829325, −18.42744229133161016419855809647, −17.476002619718658138804032055171, −17.11492881032869367800983053879, −16.104611621124421355821886974533, −15.79661140966838743317454491049, −14.67972706895205578269959675992, −13.853701208160497983374992057574, −12.63454570376637552223241057344, −12.009940486190943456022254281853, −10.88223504222095431343571044409, −10.78728210954845109906705695298, −9.65146699800790067393839893551, −8.811223074361671709645466180391, −7.9765632274429348124606685688, −6.998014374021342544686071855117, −6.34764298180944034764811205505, −5.70137277214001388098198246524, −4.70937470135990226944821982293, −3.47128844184381852745005880320, −2.0054549646269562258479226703, −1.15990968373324485944820724315, 0.05029598534599158208932324045, 1.48343250088974020124213465280, 2.121494888818342203355426453300, 3.8820081876706062529418667005, 4.171173385666539092737661081793, 5.71197627316278948904901061111, 6.56647561508693478254207592204, 7.01673128939518003962312347108, 8.230622183340343334014203465258, 9.1523039159818792921683962848, 9.71881380331114390011509868812, 10.7223613412448131222731216022, 11.28166988356152290476299192958, 11.918536461515899233042565409033, 12.69386300237165262946767734584, 13.598020651869811938545163855356, 14.99120288605124676490298879311, 15.60979574261758064258690205599, 16.48736036004387884627219987739, 17.10415964817859731706519966008, 17.6197506931173225134527578954, 18.407360316464698240839026988901, 19.20053874110646552818416143435, 19.86340846247286520316110644751, 20.80619586582354708277552964315

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