Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 6·17-s − 4·19-s + 21-s − 25-s − 27-s + 2·29-s + 4·33-s − 2·35-s − 6·37-s − 2·39-s + 2·41-s + 4·43-s + 2·45-s + 49-s + 6·51-s − 6·53-s − 8·55-s + 4·57-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-1$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \) 1.5.ac
11 \( 1 + 4 T + p T^{2} \) 1.11.e
13 \( 1 - 2 T + p T^{2} \) 1.13.ac
17 \( 1 + 6 T + p T^{2} \) 1.17.g
19 \( 1 + 4 T + p T^{2} \) 1.19.e
23 \( 1 + p T^{2} \) 1.23.a
29 \( 1 - 2 T + p T^{2} \) 1.29.ac
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 + 6 T + p T^{2} \) 1.37.g
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 + 6 T + p T^{2} \) 1.53.g
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 - 2 T + p T^{2} \) 1.61.ac
67 \( 1 + 4 T + p T^{2} \) 1.67.e
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 + 6 T + p T^{2} \) 1.73.g
79 \( 1 + 16 T + p T^{2} \) 1.79.q
83 \( 1 - 12 T + p T^{2} \) 1.83.am
89 \( 1 + 14 T + p T^{2} \) 1.89.o
97 \( 1 - 18 T + p T^{2} \) 1.97.as
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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

−9.238117136499894867345980510658, −8.535675361637489516400731752225, −7.48171223884197946068743795147, −6.47604188152629865155494776849, −6.00140300914464627775014931904, −5.08548839023881336508605977410, −4.19315915887555872024919481174, −2.78781035361267366934893473837, −1.80984559120091606425042866771, 0, 1.80984559120091606425042866771, 2.78781035361267366934893473837, 4.19315915887555872024919481174, 5.08548839023881336508605977410, 6.00140300914464627775014931904, 6.47604188152629865155494776849, 7.48171223884197946068743795147, 8.535675361637489516400731752225, 9.238117136499894867345980510658

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