Properties

Label 1-35e2-1225.242-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.765 + 0.643i$
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  + (−0.119 + 0.992i)2-s + (0.894 + 0.447i)3-s + (−0.971 − 0.237i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (0.599 + 0.800i)9-s + (−0.599 + 0.800i)11-s + (−0.762 − 0.646i)12-s + (−0.657 − 0.753i)13-s + (0.887 + 0.460i)16-s + (−0.959 − 0.280i)17-s + (−0.866 + 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.722 − 0.691i)22-s + (0.941 + 0.337i)23-s + (0.733 − 0.680i)24-s + ⋯
L(s)  = 1  + (−0.119 + 0.992i)2-s + (0.894 + 0.447i)3-s + (−0.971 − 0.237i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (0.599 + 0.800i)9-s + (−0.599 + 0.800i)11-s + (−0.762 − 0.646i)12-s + (−0.657 − 0.753i)13-s + (0.887 + 0.460i)16-s + (−0.959 − 0.280i)17-s + (−0.866 + 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.722 − 0.691i)22-s + (0.941 + 0.337i)23-s + (0.733 − 0.680i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895552062 + 0.6916071713i\)
\(L(\frac12)\) \(\approx\) \(1.895552062 + 0.6916071713i\)
\(L(1)\) \(\approx\) \(1.001344927 + 0.5885364678i\)
\(L(1)\) \(\approx\) \(1.001344927 + 0.5885364678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.119 + 0.992i)T \)
3 \( 1 + (0.894 + 0.447i)T \)
11 \( 1 + (-0.599 + 0.800i)T \)
13 \( 1 + (-0.657 - 0.753i)T \)
17 \( 1 + (-0.959 - 0.280i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.941 + 0.337i)T \)
29 \( 1 + (-0.134 - 0.990i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.762 - 0.646i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (-0.486 - 0.873i)T \)
53 \( 1 + (0.237 - 0.971i)T \)
59 \( 1 + (0.842 - 0.538i)T \)
61 \( 1 + (0.525 + 0.850i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (-0.981 - 0.193i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.512 - 0.858i)T \)
89 \( 1 + (0.946 - 0.323i)T \)
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

−20.71366828364474929076252979414, −20.079734374060291986203324150511, −19.28130355403846947371382651722, −18.87161346154418500695359041778, −18.06341514182772940776839188703, −17.34708303929370830237652723758, −16.26739813532987140581644722007, −15.29959141830511552579528152912, −14.29720506537700117135185460432, −13.79071140312294548183303830135, −13.066294120177613267844881089019, −12.361554709387946079825138561351, −11.49916229326965329871063520295, −10.678211751723219833596541915685, −9.76744581981581351120805222164, −8.937709538745443322184563936992, −8.48525941449972134282477276848, −7.47054467807501450126041850731, −6.63136180778156366800289754081, −5.20650529795033534325807243373, −4.39522552347344108981320251602, −3.251231332145640675671497303206, −2.764425951534515755523382928428, −1.756053901266065855108419851157, −0.83838519658580476540587174037, 0.457016903427381924087814299206, 2.00676621038722091347882230248, 3.02612412596969925757496819027, 4.08596225604625355743058922689, 4.92592022256179683580380304288, 5.52897789909262167660684065808, 6.993818254973126926529602224547, 7.4490195247226856769172967157, 8.26731896272149906031029784892, 9.05265897015804037804278030225, 9.91769899105888437974256748623, 10.27811718074613507711225510522, 11.682450079337663008575274170698, 13.05645892022669062144466327884, 13.27231264157371979613722263358, 14.30947125764763655133371507144, 15.0309312054687082512146881042, 15.54050761892832908963483863621, 16.11115366982216394241058481904, 17.256782269331701366471782062539, 17.76082427487500386744625840181, 18.729702108754225823225242899232, 19.40831864794595159674964615131, 20.30396649105975547384316743786, 20.8951213793165407599630851772

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