Properties

Label 2-4608-1.1-c1-0-31
Degree $2$
Conductor $4608$
Sign $-1$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.44·5-s − 4.87·7-s + 3.46·11-s + 14.7·25-s + 5.34·29-s + 10.5·31-s + 21.7·35-s + 16.7·49-s − 12.4·53-s − 15.4·55-s − 11.3·59-s − 9.79·73-s − 16.8·77-s + 3.32·79-s − 17.3·83-s − 2·97-s − 11.5·101-s + 20.2·103-s − 11.3·107-s + ⋯
L(s)  = 1  − 1.98·5-s − 1.84·7-s + 1.04·11-s + 2.95·25-s + 0.993·29-s + 1.89·31-s + 3.66·35-s + 2.39·49-s − 1.71·53-s − 2.07·55-s − 1.47·59-s − 1.14·73-s − 1.92·77-s + 0.373·79-s − 1.90·83-s − 0.203·97-s − 1.14·101-s + 1.99·103-s − 1.09·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 4.44T + 5T^{2} \)
7 \( 1 + 4.87T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.34T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 - 3.32T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−7.988967483278793011697663496352, −7.10625240620324144492886040108, −6.65349862015332969025726023516, −6.05235057756333677023197321143, −4.59896907418805351579380447249, −4.17697399318567713829293741196, −3.23957931095798611527466420214, −2.97577267528829505546894701006, −0.987705560611143327129337532796, 0, 0.987705560611143327129337532796, 2.97577267528829505546894701006, 3.23957931095798611527466420214, 4.17697399318567713829293741196, 4.59896907418805351579380447249, 6.05235057756333677023197321143, 6.65349862015332969025726023516, 7.10625240620324144492886040108, 7.988967483278793011697663496352

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