Properties

Label 2-6336-1.1-c1-0-6
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 + 4·13-s − 6·17-s + 8·19-s − 3·23-s + 4·25-s − 5·31-s + 6·35-s + 37-s − 10·43-s − 3·49-s − 6·53-s − 3·55-s − 3·59-s + 4·61-s − 12·65-s − 67-s + 15·71-s − 4·73-s − 2·77-s − 2·79-s − 6·83-s + 18·85-s + 9·89-s − 8·91-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.755·7-s + 0.301·11-s + 1.10·13-s − 1.45·17-s + 1.83·19-s − 0.625·23-s + 4/5·25-s − 0.898·31-s + 1.01·35-s + 0.164·37-s − 1.52·43-s − 3/7·49-s − 0.824·53-s − 0.404·55-s − 0.390·59-s + 0.512·61-s − 1.48·65-s − 0.122·67-s + 1.78·71-s − 0.468·73-s − 0.227·77-s − 0.225·79-s − 0.658·83-s + 1.95·85-s + 0.953·89-s − 0.838·91-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

\(L(1)\) \(\approx\) \(0.9854864139\)
\(L(\frac12)\) \(\approx\) \(0.9854864139\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−8.053710672355489314643034592341, −7.31093235292016540755307260877, −6.70956520147707782074153831968, −6.04886598194608778773456776693, −5.09532391743584753214799670039, −4.23465942787126221958004453594, −3.56855668751439570581317008792, −3.12033198305302905755997292770, −1.74926942171441723938567252118, −0.51335491910419828067793004742, 0.51335491910419828067793004742, 1.74926942171441723938567252118, 3.12033198305302905755997292770, 3.56855668751439570581317008792, 4.23465942787126221958004453594, 5.09532391743584753214799670039, 6.04886598194608778773456776693, 6.70956520147707782074153831968, 7.31093235292016540755307260877, 8.053710672355489314643034592341

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