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Properties

Label 2-336336-1.1-c1-0-166
Degree $2$
Conductor $336336$
Sign $-1$
Analytic cond. $2685.65$
Root an. cond. $51.8233$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 11-s − 13-s − 4·19-s − 5·25-s + 27-s − 10·31-s + 33-s + 2·37-s − 39-s + 6·41-s + 10·43-s + 6·53-s − 4·57-s − 2·61-s − 2·67-s − 12·71-s + 10·73-s − 5·75-s + 10·79-s + 81-s − 12·83-s − 12·89-s − 10·93-s + 10·97-s + 99-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.917·19-s − 25-s + 0.192·27-s − 1.79·31-s + 0.174·33-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.52·43-s + 0.824·53-s − 0.529·57-s − 0.256·61-s − 0.244·67-s − 1.42·71-s + 1.17·73-s − 0.577·75-s + 1.12·79-s + 1/9·81-s − 1.31·83-s − 1.27·89-s − 1.03·93-s + 1.01·97-s + 0.100·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336336\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2685.65\)
Root analytic conductor: \(51.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 336336,\ (\ :1/2),\ -1)\)

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

−12.71345620563763, −12.57157298253557, −11.92329051746097, −11.43181976871249, −10.95469487618054, −10.55887172755781, −10.07165061776774, −9.448379309296912, −9.200873511241277, −8.798812222093735, −8.180557576896116, −7.750675699693663, −7.339638288961945, −6.883287455459361, −6.308299530938682, −5.656939525646050, −5.524524091493215, −4.517550849357220, −4.261916975770371, −3.776998736610550, −3.209496909548886, −2.522504653383295, −2.108464182285687, −1.575098568290454, −0.7712851556694101, 0, 0.7712851556694101, 1.575098568290454, 2.108464182285687, 2.522504653383295, 3.209496909548886, 3.776998736610550, 4.261916975770371, 4.517550849357220, 5.524524091493215, 5.656939525646050, 6.308299530938682, 6.883287455459361, 7.339638288961945, 7.750675699693663, 8.180557576896116, 8.798812222093735, 9.200873511241277, 9.448379309296912, 10.07165061776774, 10.55887172755781, 10.95469487618054, 11.43181976871249, 11.92329051746097, 12.57157298253557, 12.71345620563763

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