Properties

Label 2-336-7.2-c5-0-25
Degree $2$
Conductor $336$
Sign $0.944 - 0.328i$
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 + (35.0 − 60.7i)5-s + (90.7 − 92.5i)7-s + (−40.5 + 70.1i)9-s + (311. + 539. i)11-s − 278.·13-s + 631.·15-s + (443. + 768. i)17-s + (−791. + 1.37e3i)19-s + (1.12e3 + 290. i)21-s + (−969. + 1.67e3i)23-s + (−898. − 1.55e3i)25-s − 729·27-s + 8.57e3·29-s + (1.65e3 + 2.86e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.627 − 1.08i)5-s + (0.700 − 0.714i)7-s + (−0.166 + 0.288i)9-s + (0.776 + 1.34i)11-s − 0.457·13-s + 0.724·15-s + (0.372 + 0.644i)17-s + (−0.503 + 0.871i)19-s + (0.559 + 0.143i)21-s + (−0.382 + 0.662i)23-s + (−0.287 − 0.497i)25-s − 0.192·27-s + 1.89·29-s + (0.309 + 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.944 - 0.328i$
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: \((2,\ 336,\ (\ :5/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.034774548\)
\(L(\frac12)\) \(\approx\) \(3.034774548\)
\(L(\frac{7}{2})\) 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 \( 1 + (-90.7 + 92.5i)T \)
good5 \( 1 + (-35.0 + 60.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-311. - 539. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 278.T + 3.71e5T^{2} \)
17 \( 1 + (-443. - 768. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (791. - 1.37e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (969. - 1.67e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 8.57e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.65e3 - 2.86e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-6.82e3 + 1.18e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.51e3T + 1.15e8T^{2} \)
43 \( 1 + 1.99e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.26e4 + 2.18e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (584. + 1.01e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.35e4 - 2.34e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (6.97e3 - 1.20e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.00e4 - 3.47e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.31e4T + 1.80e9T^{2} \)
73 \( 1 + (-4.12e4 - 7.14e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-5.07e4 + 8.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.93e4T + 3.93e9T^{2} \)
89 \( 1 + (46.7 - 80.9i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.45e4T + 8.58e9T^{2} \)
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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

−10.38529354777923861483576974964, −9.942880012667591083030172008345, −8.949180640511188018007874874073, −8.116616883994873734938203989192, −7.04328287876752481107449892252, −5.63946621835542722415577334234, −4.62529790903727187301106449725, −3.99185971352969124583512564871, −2.04449539644799475351135368614, −1.13756891098593760465010317129, 0.892600307170863103220618671815, 2.38752030129742490123908830929, 3.01718323650275974042980949288, 4.74702855361792152334002957247, 6.15555462506638874127916711410, 6.57349986774754397251873465979, 7.913906535272166821284544794493, 8.715349504953744118781476423287, 9.694134703308338130911884328093, 10.80889134184215690837375911274

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