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Properties

Label 2-204e2-1.1-c1-0-15
Degree $2$
Conductor $41616$
Sign $-1$
Analytic cond. $332.305$
Root an. cond. $18.2292$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 4·7-s − 3·11-s + 2·13-s − 8·19-s − 6·23-s + 4·25-s + 3·29-s − 7·31-s + 12·35-s − 8·37-s + 6·41-s + 4·43-s − 6·47-s + 9·49-s + 9·53-s + 9·55-s + 15·59-s − 14·61-s − 6·65-s − 2·67-s + 7·73-s + 12·77-s − 79-s + 12·83-s − 8·91-s + 24·95-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.51·7-s − 0.904·11-s + 0.554·13-s − 1.83·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s − 1.25·31-s + 2.02·35-s − 1.31·37-s + 0.937·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s + 1.23·53-s + 1.21·55-s + 1.95·59-s − 1.79·61-s − 0.744·65-s − 0.244·67-s + 0.819·73-s + 1.36·77-s − 0.112·79-s + 1.31·83-s − 0.838·91-s + 2.46·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41616\)    =    \(2^{4} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(332.305\)
Root analytic conductor: \(18.2292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 41616,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.09238949299453, −14.64362952573817, −13.88710829122633, −13.26260074371229, −12.90907017672780, −12.36462777454726, −12.09793547941877, −11.29808569261700, −10.76043234818288, −10.35953844205215, −9.878149366529622, −8.965757979371691, −8.715824385050869, −7.991887072390373, −7.602186300528041, −6.897808622811558, −6.394994613845959, −5.889825598871511, −5.140833134241631, −4.259879179337576, −3.844746519957863, −3.417867219367989, −2.615562617848843, −1.953608942919883, −0.5519007626239684, 0, 0.5519007626239684, 1.953608942919883, 2.615562617848843, 3.417867219367989, 3.844746519957863, 4.259879179337576, 5.140833134241631, 5.889825598871511, 6.394994613845959, 6.897808622811558, 7.602186300528041, 7.991887072390373, 8.715824385050869, 8.965757979371691, 9.878149366529622, 10.35953844205215, 10.76043234818288, 11.29808569261700, 12.09793547941877, 12.36462777454726, 12.90907017672780, 13.26260074371229, 13.88710829122633, 14.64362952573817, 15.09238949299453

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