Properties

Label 1-35e2-1225.333-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.627 - 0.778i$
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  + (0.323 + 0.946i)2-s + (0.635 − 0.772i)3-s + (−0.791 + 0.611i)4-s + (0.936 + 0.351i)6-s + (−0.834 − 0.550i)8-s + (−0.193 − 0.981i)9-s + (0.193 − 0.981i)11-s + (−0.0299 + 0.999i)12-s + (−0.919 − 0.393i)13-s + (0.251 − 0.967i)16-s + (0.379 + 0.925i)17-s + (0.866 − 0.5i)18-s + (0.978 + 0.207i)19-s + (0.990 − 0.134i)22-s + (−0.850 + 0.525i)23-s + (−0.955 + 0.294i)24-s + ⋯
L(s)  = 1  + (0.323 + 0.946i)2-s + (0.635 − 0.772i)3-s + (−0.791 + 0.611i)4-s + (0.936 + 0.351i)6-s + (−0.834 − 0.550i)8-s + (−0.193 − 0.981i)9-s + (0.193 − 0.981i)11-s + (−0.0299 + 0.999i)12-s + (−0.919 − 0.393i)13-s + (0.251 − 0.967i)16-s + (0.379 + 0.925i)17-s + (0.866 − 0.5i)18-s + (0.978 + 0.207i)19-s + (0.990 − 0.134i)22-s + (−0.850 + 0.525i)23-s + (−0.955 + 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.627 - 0.778i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3960672440 - 0.8283212217i\)
\(L(\frac12)\) \(\approx\) \(0.3960672440 - 0.8283212217i\)
\(L(1)\) \(\approx\) \(1.170016663 + 0.1128758751i\)
\(L(1)\) \(\approx\) \(1.170016663 + 0.1128758751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.323 + 0.946i)T \)
3 \( 1 + (0.635 - 0.772i)T \)
11 \( 1 + (0.193 - 0.981i)T \)
13 \( 1 + (-0.919 - 0.393i)T \)
17 \( 1 + (0.379 + 0.925i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (-0.850 + 0.525i)T \)
29 \( 1 + (0.691 - 0.722i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.0299 + 0.999i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.817 - 0.575i)T \)
53 \( 1 + (0.611 + 0.791i)T \)
59 \( 1 + (-0.712 - 0.701i)T \)
61 \( 1 + (-0.337 + 0.941i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.800 - 0.599i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.0896 - 0.995i)T \)
89 \( 1 + (-0.992 - 0.119i)T \)
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

−21.17093634610309023089630785780, −20.3974074186469171093494419906, −19.88127176817021168923710775111, −19.32286149950793255747812193354, −18.23938454864479115945949427082, −17.62693905156101563092767691068, −16.45335835966831486352489935827, −15.71651054777971062557878017969, −14.71915199659730813882464119250, −14.258149014353019173764725055461, −13.63284891174279156945461936048, −12.44583600540716866137839739358, −11.95828122969241567922983394721, −10.979800538528600553790299604844, −10.014909242369049664325244641675, −9.67488865899289195092568356632, −8.9246935935710121298409413070, −7.87277375663591915757814583013, −6.89042437227093472376519194299, −5.42010566335920900879039202721, −4.75506480822312030203712859485, −4.13380434114215123896803207731, −3.01252804571282069732966418852, −2.449436151085000209756278346453, −1.36212769064136940864617280463, 0.14833995678573541742360583771, 1.25752090485126096302270593633, 2.708736684226120481251151497329, 3.43121647765826657684231962615, 4.38040465871893258215008799519, 5.70074423000531135746767598498, 6.11922045092111180268293821538, 7.184102927902649172635791857753, 7.91171454074192928297423839468, 8.37295847448100882575320271100, 9.380805340382123247159065129783, 10.14032596542857115449103067464, 11.77179048468134712807824477596, 12.18948496281544781883109993254, 13.217358122150637920910018327657, 13.76796273630511154608671056504, 14.40990315963664597373630473124, 15.166122551306277147091819600098, 15.90178595028568355918677132964, 16.934581774976783582601383946233, 17.48529725933024243041629061920, 18.31425516355701363015109107322, 19.08797617769665571401412221290, 19.68971076312145206906310022731, 20.73774859478661556441945070307

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