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Properties

Label 2-55e2-275.203-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.962 - 0.269i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
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  + (0.412 − 0.809i)3-s + i·4-s + (−0.951 − 0.309i)5-s + (0.103 + 0.142i)9-s + (0.809 + 0.412i)12-s + (−0.642 + 0.642i)15-s − 16-s + (0.309 − 0.951i)20-s + (0.278 − 0.142i)23-s + (0.809 + 0.587i)25-s + (1.05 − 0.166i)27-s + (1.53 + 1.11i)31-s + (−0.142 + 0.103i)36-s + (1.39 − 0.221i)37-s + (−0.0542 − 0.166i)45-s + ⋯
L(s)  = 1  + (0.412 − 0.809i)3-s + i·4-s + (−0.951 − 0.309i)5-s + (0.103 + 0.142i)9-s + (0.809 + 0.412i)12-s + (−0.642 + 0.642i)15-s − 16-s + (0.309 − 0.951i)20-s + (0.278 − 0.142i)23-s + (0.809 + 0.587i)25-s + (1.05 − 0.166i)27-s + (1.53 + 1.11i)31-s + (−0.142 + 0.103i)36-s + (1.39 − 0.221i)37-s + (−0.0542 − 0.166i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226218125\)
\(L(\frac12)\) \(\approx\) \(1.226218125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-0.412 + 0.809i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (1.58 - 0.809i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.686933226287984831336921236901, −8.022749406967939951492648960129, −7.59146597436678545065549519824, −7.01758382111560481625713143716, −6.15159905811232806703694243753, −4.71839702517895848881466922277, −4.30695188306638627751142054175, −3.16553286509764407388887652796, −2.56944564884848033712514978514, −1.18112531307190446874287159353, 0.884641997395090119349152948917, 2.43974566338479374835069209375, 3.37300934457403240805347809202, 4.31836705048384032035984389996, 4.70007615592998940227279274966, 5.84513595218994614470444014605, 6.58235698119619460818367204178, 7.36582457587532824747424686723, 8.270853582526218989036076211788, 8.937699171401110361215437667237

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