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Properties

Label 1-35e2-1225.387-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.577 - 0.816i$
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  + (−0.817 + 0.575i)2-s + (−0.538 + 0.842i)3-s + (0.337 − 0.941i)4-s + (−0.0448 − 0.998i)6-s + (0.266 + 0.963i)8-s + (−0.420 − 0.907i)9-s + (0.420 − 0.907i)11-s + (0.611 + 0.791i)12-s + (0.512 − 0.858i)13-s + (−0.772 − 0.635i)16-s + (−0.762 − 0.646i)17-s + (0.866 + 0.5i)18-s + (0.104 − 0.994i)19-s + (0.178 + 0.983i)22-s + (0.379 + 0.925i)23-s + (−0.955 − 0.294i)24-s + ⋯
L(s)  = 1  + (−0.817 + 0.575i)2-s + (−0.538 + 0.842i)3-s + (0.337 − 0.941i)4-s + (−0.0448 − 0.998i)6-s + (0.266 + 0.963i)8-s + (−0.420 − 0.907i)9-s + (0.420 − 0.907i)11-s + (0.611 + 0.791i)12-s + (0.512 − 0.858i)13-s + (−0.772 − 0.635i)16-s + (−0.762 − 0.646i)17-s + (0.866 + 0.5i)18-s + (0.104 − 0.994i)19-s + (0.178 + 0.983i)22-s + (0.379 + 0.925i)23-s + (−0.955 − 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7746204755 - 0.4007553283i\)
\(L(\frac12)\) \(\approx\) \(0.7746204755 - 0.4007553283i\)
\(L(1)\) \(\approx\) \(0.6212349260 + 0.1274653999i\)
\(L(1)\) \(\approx\) \(0.6212349260 + 0.1274653999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.817 + 0.575i)T \)
3 \( 1 + (-0.538 + 0.842i)T \)
11 \( 1 + (0.420 - 0.907i)T \)
13 \( 1 + (0.512 - 0.858i)T \)
17 \( 1 + (-0.762 - 0.646i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.379 + 0.925i)T \)
29 \( 1 + (-0.473 + 0.880i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.611 + 0.791i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (-0.800 - 0.599i)T \)
53 \( 1 + (0.941 + 0.337i)T \)
59 \( 1 + (0.163 - 0.986i)T \)
61 \( 1 + (-0.280 + 0.959i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.999 - 0.0149i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.919 + 0.393i)T \)
89 \( 1 + (0.873 + 0.486i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−20.974651187793259724139652553477, −20.039734525716469312045237815, −19.43093206765936772168037919498, −18.69156419497264451403362297568, −18.10522277668490097409174370535, −17.36017273467052653993003870382, −16.757929179654139109785943767184, −16.05550690814743419494702509975, −14.895370975415549114033621127708, −13.857015675629867090748095102436, −12.94094166138705391642497299878, −12.39254005001275341396259703436, −11.61509706933014293145814527012, −10.995219079109744315018623801119, −10.1230529170979788540284193293, −9.22693985235837238966741771804, −8.32326010651757371661670559966, −7.64499508397466578786379928395, −6.62031547426515144616982708236, −6.26128295124505355226521993637, −4.6792427205336912076447355971, −3.86991323578015639948257341343, −2.45467735773822224310711530788, −1.786487988968088787445243287582, −0.94162141384222129019243364077, 0.33771827909978002380033992188, 1.05323874281180050876757262477, 2.67292953147495465913033645713, 3.69003596907459152458295669197, 4.930045855714258786255237833922, 5.53652163688341716071333835878, 6.39033120946599191930819065885, 7.14942760667487231400589211597, 8.36604207900249221496081745665, 8.980507342465750786181056745765, 9.66705775716109331757579794645, 10.613177675592326791306724345903, 11.19879494881214503489012236502, 11.766149214158553049165095016287, 13.27346022981651253811782514260, 14.004198014540267737166732048356, 15.15623230661721386595900217674, 15.502730308548129739267539863682, 16.24906682041858688420058831768, 16.98203656174677852839027429598, 17.64281102545651069699937718335, 18.26299267486908533715135517643, 19.22667783595210926551942358260, 20.08334591515647215594284663820, 20.63251373439627548004466217503

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