Properties

Label 2-336-12.11-c3-0-26
Degree $2$
Conductor $336$
Sign $-0.907 + 0.420i$
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  + (−2.18 − 4.71i)3-s + 14.5i·5-s + 7i·7-s + (−17.4 + 20.6i)9-s + 31.0·11-s − 55.7·13-s + (68.7 − 31.8i)15-s − 86.4i·17-s − 96.8i·19-s + (32.9 − 15.3i)21-s − 115.·23-s − 87.7·25-s + (135. + 37.0i)27-s − 49.1i·29-s + 12.5i·31-s + ⋯
L(s)  = 1  + (−0.420 − 0.907i)3-s + 1.30i·5-s + 0.377i·7-s + (−0.645 + 0.763i)9-s + 0.850·11-s − 1.18·13-s + (1.18 − 0.549i)15-s − 1.23i·17-s − 1.16i·19-s + (0.342 − 0.159i)21-s − 1.04·23-s − 0.701·25-s + (0.964 + 0.264i)27-s − 0.314i·29-s + 0.0728i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3590024833\)
\(L(\frac12)\) \(\approx\) \(0.3590024833\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.18 + 4.71i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 14.5iT - 125T^{2} \)
11 \( 1 - 31.0T + 1.33e3T^{2} \)
13 \( 1 + 55.7T + 2.19e3T^{2} \)
17 \( 1 + 86.4iT - 4.91e3T^{2} \)
19 \( 1 + 96.8iT - 6.85e3T^{2} \)
23 \( 1 + 115.T + 1.21e4T^{2} \)
29 \( 1 + 49.1iT - 2.43e4T^{2} \)
31 \( 1 - 12.5iT - 2.97e4T^{2} \)
37 \( 1 + 296.T + 5.06e4T^{2} \)
41 \( 1 + 213. iT - 6.89e4T^{2} \)
43 \( 1 + 165. iT - 7.95e4T^{2} \)
47 \( 1 + 426.T + 1.03e5T^{2} \)
53 \( 1 - 460. iT - 1.48e5T^{2} \)
59 \( 1 - 686.T + 2.05e5T^{2} \)
61 \( 1 + 583.T + 2.26e5T^{2} \)
67 \( 1 + 589. iT - 3.00e5T^{2} \)
71 \( 1 + 766.T + 3.57e5T^{2} \)
73 \( 1 + 904.T + 3.89e5T^{2} \)
79 \( 1 + 459. iT - 4.93e5T^{2} \)
83 \( 1 - 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 119. iT - 7.04e5T^{2} \)
97 \( 1 + 331.T + 9.12e5T^{2} \)
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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

−10.92100434268557769368271653745, −9.914140905154905997314861710942, −8.824363586717863417593588683699, −7.39366687817227197513467883586, −7.00833231664581426321766845508, −6.06435328210469012427010420071, −4.85024385958063464793572545478, −3.04101459571552117070506070340, −2.10015595523158780743665878514, −0.13300623891011435106979809880, 1.49141872216298845711340809048, 3.69017829664116082319157689658, 4.49449899100874662068413442762, 5.42812009220802401719800742247, 6.46685691392779426109626789950, 8.010289552057759633360722394446, 8.827281904706021419465158713550, 9.799233938111713065340995012145, 10.34136011476257316789157574348, 11.70136759238883282744698777139

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