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Properties

Label 2-336-84.11-c3-0-43
Degree $2$
Conductor $336$
Sign $0.606 + 0.794i$
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  + (5.04 − 1.24i)3-s + (11.5 − 6.67i)5-s + (16.4 − 8.45i)7-s + (23.9 − 12.5i)9-s + (11.6 − 20.1i)11-s − 10.0·13-s + (50.0 − 48.0i)15-s + (−11.8 − 6.83i)17-s + (−86.0 + 49.6i)19-s + (72.6 − 63.1i)21-s + (19.9 + 34.6i)23-s + (26.7 − 46.3i)25-s + (105. − 92.9i)27-s + 107. i·29-s + (−190. − 109. i)31-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)3-s + (1.03 − 0.597i)5-s + (0.889 − 0.456i)7-s + (0.885 − 0.464i)9-s + (0.318 − 0.551i)11-s − 0.213·13-s + (0.861 − 0.827i)15-s + (−0.168 − 0.0974i)17-s + (−1.03 + 0.599i)19-s + (0.754 − 0.655i)21-s + (0.181 + 0.314i)23-s + (0.213 − 0.370i)25-s + (0.748 − 0.662i)27-s + 0.686i·29-s + (−1.10 − 0.635i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.606 + 0.794i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.606 + 0.794i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.451085034\)
\(L(\frac12)\) \(\approx\) \(3.451085034\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.04 + 1.24i)T \)
7 \( 1 + (-16.4 + 8.45i)T \)
good5 \( 1 + (-11.5 + 6.67i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-11.6 + 20.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 10.0T + 2.19e3T^{2} \)
17 \( 1 + (11.8 + 6.83i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (86.0 - 49.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-19.9 - 34.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 107. iT - 2.43e4T^{2} \)
31 \( 1 + (190. + 109. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-36.4 - 63.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 469. iT - 6.89e4T^{2} \)
43 \( 1 + 223. iT - 7.95e4T^{2} \)
47 \( 1 + (29.2 + 50.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-152. - 88.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-302. + 523. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-452. - 784. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (536. + 310. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 415.T + 3.57e5T^{2} \)
73 \( 1 + (80.5 - 139. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-25.4 + 14.7i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + (-1.16e3 + 674. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 229.T + 9.12e5T^{2} \)
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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

−10.85442119734819955724660969043, −9.886915559630839513869650346017, −9.007896225823588746305840501085, −8.325720051955133677183164241999, −7.32136087098640536181558072612, −6.12522565360744777735577533765, −4.91810995997862792977511324846, −3.75280734432503269754359918764, −2.16300051779223048190078453230, −1.24061843907061468502520337379, 1.87201014790967245281563477584, 2.51519468940711060817953142548, 4.11041594829903800716884785994, 5.22287254851915601206277286217, 6.52418155223834486106489329543, 7.50604826455568500693245290961, 8.649782862938722887972343482009, 9.299981333019234135441510905480, 10.28347957436061478164753514660, 10.97629045262480672068310969699

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