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Properties

Label 2-336-21.5-c3-0-31
Degree $2$
Conductor $336$
Sign $0.766 + 0.642i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
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 − 2.59i)3-s + (8.5 + 16.4i)7-s + (13.5 − 23.3i)9-s − 91.7i·13-s + (109.5 + 63.2i)19-s + (81 + 51.9i)21-s + (62.5 + 108. i)25-s − 140. i·27-s + (163.5 − 94.3i)31-s + (161.5 − 279. i)37-s + (−238.5 − 413. i)39-s + 71·43-s + (−198.5 + 279. i)49-s + 657·57-s + (−810 − 467. i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.458 + 0.888i)7-s + (0.5 − 0.866i)9-s − 1.95i·13-s + (1.32 + 0.763i)19-s + (0.841 + 0.539i)21-s + (0.5 + 0.866i)25-s − 1.00i·27-s + (0.947 − 0.546i)31-s + (0.717 − 1.24i)37-s + (−0.979 − 1.69i)39-s + 0.251·43-s + (−0.578 + 0.815i)49-s + 1.52·57-s + (−1.70 − 0.981i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.772041143\)
\(L(\frac12)\) \(\approx\) \(2.772041143\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 + 2.59i)T \)
7 \( 1 + (-8.5 - 16.4i)T \)
good5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 91.7iT - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-109.5 - 63.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-163.5 + 94.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-161.5 + 279. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 71T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (810 + 467. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (63.5 + 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-1.05e3 + 608. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (693.5 - 1.20e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.37e3iT - 9.12e5T^{2} \)
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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

−11.06537468872563047691192576599, −9.898436270261964888646303737860, −9.074455748968271670387811815193, −7.999309970429956483031113772457, −7.62543310731963777447804636875, −6.08387713149217653397674176283, −5.17658193791306964161874585166, −3.46926703289478087203484688916, −2.54891026651341968682770360274, −1.06303868262728222591978223276, 1.40197601495288475041682825195, 2.86830823917034559603212827896, 4.22052943577324992919404934687, 4.80677589142445616698285578104, 6.65433313606475283522592152497, 7.47380754882913727864428415743, 8.509991046590564321369141419154, 9.372328076866202656009549970451, 10.15644293581712758957229695857, 11.16746101750626670069516557682

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