Properties

Label 2-6336-1.1-c1-0-23
Degree $2$
Conductor $6336$
Sign $1$
Analytic cond. $50.5932$
Root an. cond. $7.11289$
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  − 3·5-s + 2·7-s + 11-s + 6·17-s + 4·19-s + 23-s + 4·25-s − 8·29-s + 7·31-s − 6·35-s + 37-s − 4·41-s + 6·43-s − 8·47-s − 3·49-s + 2·53-s − 3·55-s + 59-s − 4·61-s − 5·67-s + 3·71-s + 16·73-s + 2·77-s − 2·79-s + 2·83-s − 18·85-s − 15·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.755·7-s + 0.301·11-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 4/5·25-s − 1.48·29-s + 1.25·31-s − 1.01·35-s + 0.164·37-s − 0.624·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.404·55-s + 0.130·59-s − 0.512·61-s − 0.610·67-s + 0.356·71-s + 1.87·73-s + 0.227·77-s − 0.225·79-s + 0.219·83-s − 1.95·85-s − 1.58·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(50.5932\)
Root analytic conductor: \(7.11289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736087912\)
\(L(\frac12)\) \(\approx\) \(1.736087912\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.927629123853046774697751456251, −7.55252528081176974112132450170, −6.82671507782272554192070208796, −5.79242088578546656641015440150, −5.10583765748157113159455391169, −4.37221035310191883411362450637, −3.60822760858280223727828451956, −3.02374111947685053319903637317, −1.66294893612641948046386286042, −0.72016671080523259380610994522, 0.72016671080523259380610994522, 1.66294893612641948046386286042, 3.02374111947685053319903637317, 3.60822760858280223727828451956, 4.37221035310191883411362450637, 5.10583765748157113159455391169, 5.79242088578546656641015440150, 6.82671507782272554192070208796, 7.55252528081176974112132450170, 7.927629123853046774697751456251

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