Properties

Label 2-3648-1.1-c1-0-25
Degree $2$
Conductor $3648$
Sign $1$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 4·7-s + 9-s + 6·11-s + 2·13-s + 2·15-s + 6·17-s − 19-s − 4·21-s + 6·23-s − 25-s − 27-s − 4·29-s − 6·33-s − 8·35-s + 6·37-s − 2·39-s + 12·41-s − 2·45-s − 2·47-s + 9·49-s − 6·51-s − 12·53-s − 12·55-s + 57-s + 8·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.04·33-s − 1.35·35-s + 0.986·37-s − 0.320·39-s + 1.87·41-s − 0.298·45-s − 0.291·47-s + 9/7·49-s − 0.840·51-s − 1.64·53-s − 1.61·55-s + 0.132·57-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(29.1294\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033584150\)
\(L(\frac12)\) \(\approx\) \(2.033584150\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.501994221085784506502817541541, −7.63870668650087109482054051645, −7.30477523417000272051173704467, −6.19826130044277810938742089352, −5.59150094429109137122777969674, −4.52452987582281542071008706793, −4.15217436255771555307261745962, −3.21637007880483542993719877178, −1.57910401514899058156295584774, −0.994467879463985454107690875367, 0.994467879463985454107690875367, 1.57910401514899058156295584774, 3.21637007880483542993719877178, 4.15217436255771555307261745962, 4.52452987582281542071008706793, 5.59150094429109137122777969674, 6.19826130044277810938742089352, 7.30477523417000272051173704467, 7.63870668650087109482054051645, 8.501994221085784506502817541541

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