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Properties

Label 2-336-7.4-c5-0-33
Degree $2$
Conductor $336$
Sign $-0.602 + 0.797i$
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 + (−33.1 − 57.3i)5-s + (115. − 57.9i)7-s + (−40.5 − 70.1i)9-s + (376. − 652. i)11-s + 484.·13-s − 596.·15-s + (541. − 938. i)17-s + (874. + 1.51e3i)19-s + (70.2 − 1.16e3i)21-s + (−86.2 − 149. i)23-s + (−631. + 1.09e3i)25-s − 729·27-s + 8.31e3·29-s + (−748. + 1.29e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.592 − 1.02i)5-s + (0.894 − 0.446i)7-s + (−0.166 − 0.288i)9-s + (0.939 − 1.62i)11-s + 0.795·13-s − 0.684·15-s + (0.454 − 0.787i)17-s + (0.556 + 0.963i)19-s + (0.0347 − 0.576i)21-s + (−0.0339 − 0.0588i)23-s + (−0.202 + 0.350i)25-s − 0.192·27-s + 1.83·29-s + (−0.139 + 0.242i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.602 + 0.797i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.602 + 0.797i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.681031593\)
\(L(\frac12)\) \(\approx\) \(2.681031593\)
\(L(\frac{7}{2})\) 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 \( 1 + (-115. + 57.9i)T \)
good5 \( 1 + (33.1 + 57.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-376. + 652. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 484.T + 3.71e5T^{2} \)
17 \( 1 + (-541. + 938. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-874. - 1.51e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (86.2 + 149. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 8.31e3T + 2.05e7T^{2} \)
31 \( 1 + (748. - 1.29e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.03e3 - 1.78e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 8.32e3T + 1.15e8T^{2} \)
43 \( 1 + 7.13e3T + 1.47e8T^{2} \)
47 \( 1 + (6.89e3 + 1.19e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (6.72e3 - 1.16e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.03e3 - 5.25e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (8.17e3 + 1.41e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.79e4 - 4.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 5.98e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.77e3 + 6.53e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.40e4 + 5.90e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.09e5T + 3.93e9T^{2} \)
89 \( 1 + (-4.58e4 - 7.93e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.32e4T + 8.58e9T^{2} \)
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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

−10.53038402184192751757691662521, −9.098161513721601503587048288900, −8.369682459390150403164654594086, −7.895321693970197385678671702437, −6.54834918969682556941674837873, −5.42846864355294765970049866071, −4.23829419371389018410295384270, −3.24597550013706458922945365021, −1.24616895834780445869865780884, −0.820076142961677572401618159923, 1.50327646556792717883312882957, 2.81863520244851108987865687231, 3.99681610739682097715330223432, 4.85824940209510647545164375298, 6.34800522052923651877386801758, 7.30731952007571906959706773844, 8.207820378350387656174702379815, 9.219662919411563804054414406414, 10.17845323498853371991313571195, 11.09603178665596184805114333459

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