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Properties

Label 2-336-7.2-c5-0-19
Degree $2$
Conductor $336$
Sign $0.586 - 0.810i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 + 7.79i)3-s + (−29.1 + 50.5i)5-s + (21.4 − 127. i)7-s + (−40.5 + 70.1i)9-s + (8.71 + 15.0i)11-s + 889.·13-s − 525.·15-s + (−513. − 889. i)17-s + (−869. + 1.50e3i)19-s + (1.09e3 − 408. i)21-s + (1.96e3 − 3.40e3i)23-s + (−140. − 243. i)25-s − 729·27-s + 5.63e3·29-s + (−1.54e3 − 2.68e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.522 + 0.904i)5-s + (0.165 − 0.986i)7-s + (−0.166 + 0.288i)9-s + (0.0217 + 0.0376i)11-s + 1.46·13-s − 0.602·15-s + (−0.430 − 0.746i)17-s + (−0.552 + 0.957i)19-s + (0.540 − 0.202i)21-s + (0.775 − 1.34i)23-s + (−0.0450 − 0.0780i)25-s − 0.192·27-s + 1.24·29-s + (−0.289 − 0.501i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.196029140\)
\(L(\frac12)\) \(\approx\) \(2.196029140\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 + (-21.4 + 127. i)T \)
good5 \( 1 + (29.1 - 50.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-8.71 - 15.0i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 889.T + 3.71e5T^{2} \)
17 \( 1 + (513. + 889. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (869. - 1.50e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.96e3 + 3.40e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 5.63e3T + 2.05e7T^{2} \)
31 \( 1 + (1.54e3 + 2.68e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.51e3 - 4.35e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.83e4T + 1.15e8T^{2} \)
43 \( 1 - 1.63e3T + 1.47e8T^{2} \)
47 \( 1 + (4.80e3 - 8.31e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.16e4 - 2.01e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.80e3 + 3.12e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.14e4 - 1.98e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.35e4 - 4.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.59e3T + 1.80e9T^{2} \)
73 \( 1 + (2.96e3 + 5.13e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.42e4 - 7.66e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 9.58e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.32e4 + 4.02e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 7.59e4T + 8.58e9T^{2} \)
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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

−10.81845552801318497260533158376, −10.22052361262340897865217656962, −8.924813441149552641339695223021, −8.047838927261882771283418448276, −7.06118750292807727269279951565, −6.17944106269666042499991005988, −4.53596664781489253920356438652, −3.77621961891688628790151425506, −2.72614166914596173409571298870, −0.933731316837381460171144728311, 0.73713304501588895031437813525, 1.89160443187907557869238342379, 3.32059230913794734718878888812, 4.56773386936454397152918161929, 5.70824697715001969805306874125, 6.69556388592209961692825129289, 8.034178751963667908387590244612, 8.700624495102366445353768188134, 9.175952580761879908177803536265, 10.84447073459460148088349653289

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