Properties

Label 2-336-1.1-c3-0-9
Degree $2$
Conductor $336$
Sign $1$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 18·5-s − 7·7-s + 9·9-s + 72·11-s − 34·13-s + 54·15-s + 6·17-s − 92·19-s − 21·21-s + 180·23-s + 199·25-s + 27·27-s − 114·29-s − 56·31-s + 216·33-s − 126·35-s − 34·37-s − 102·39-s + 6·41-s − 164·43-s + 162·45-s − 168·47-s + 49·49-s + 18·51-s + 654·53-s + 1.29e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.60·5-s − 0.377·7-s + 1/3·9-s + 1.97·11-s − 0.725·13-s + 0.929·15-s + 0.0856·17-s − 1.11·19-s − 0.218·21-s + 1.63·23-s + 1.59·25-s + 0.192·27-s − 0.729·29-s − 0.324·31-s + 1.13·33-s − 0.608·35-s − 0.151·37-s − 0.418·39-s + 0.0228·41-s − 0.581·43-s + 0.536·45-s − 0.521·47-s + 1/7·49-s + 0.0494·51-s + 1.69·53-s + 3.17·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{336} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.167690738\)
\(L(\frac12)\) \(\approx\) \(3.167690738\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
7 \( 1 + p T \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 72 T + p^{3} T^{2} \)
13 \( 1 + 34 T + p^{3} T^{2} \)
17 \( 1 - 6 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 - 180 T + p^{3} T^{2} \)
29 \( 1 + 114 T + p^{3} T^{2} \)
31 \( 1 + 56 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 - 6 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 + 168 T + p^{3} T^{2} \)
53 \( 1 - 654 T + p^{3} T^{2} \)
59 \( 1 - 492 T + p^{3} T^{2} \)
61 \( 1 + 250 T + p^{3} T^{2} \)
67 \( 1 - 124 T + p^{3} T^{2} \)
71 \( 1 + 36 T + p^{3} T^{2} \)
73 \( 1 - 1010 T + p^{3} T^{2} \)
79 \( 1 + 56 T + p^{3} T^{2} \)
83 \( 1 + 228 T + p^{3} T^{2} \)
89 \( 1 - 390 T + p^{3} T^{2} \)
97 \( 1 + 70 T + p^{3} T^{2} \)
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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

−10.97063371379144635756312674025, −9.887411038356644898325913128793, −9.309185910214281596244060836286, −8.704711987937490937472325150768, −6.98841003189115270424790114680, −6.44031836747561565089771180367, −5.23966758715370911869091302667, −3.85577483848919244336909269014, −2.46611696453234412923932194191, −1.37817848240816738073129413396, 1.37817848240816738073129413396, 2.46611696453234412923932194191, 3.85577483848919244336909269014, 5.23966758715370911869091302667, 6.44031836747561565089771180367, 6.98841003189115270424790114680, 8.704711987937490937472325150768, 9.309185910214281596244060836286, 9.887411038356644898325913128793, 10.97063371379144635756312674025

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