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Properties

Label 2-336-7.5-c6-0-10
Degree $2$
Conductor $336$
Sign $0.823 - 0.567i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 − 7.79i)3-s + (−68.1 − 39.3i)5-s + (−335. + 70.2i)7-s + (121.5 − 210. i)9-s + (−457. − 791. i)11-s − 87.6i·13-s − 1.22e3·15-s + (−6.25e3 + 3.61e3i)17-s + (3.88e3 + 2.24e3i)19-s + (−3.98e3 + 3.56e3i)21-s + (3.46e3 − 6.00e3i)23-s + (−4.72e3 − 8.17e3i)25-s − 3.78e3i·27-s − 1.34e4·29-s + (2.87e4 − 1.66e4i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.544 − 0.314i)5-s + (−0.978 + 0.204i)7-s + (0.166 − 0.288i)9-s + (−0.343 − 0.594i)11-s − 0.0398i·13-s − 0.363·15-s + (−1.27 + 0.735i)17-s + (0.566 + 0.326i)19-s + (−0.430 + 0.385i)21-s + (0.285 − 0.493i)23-s + (−0.302 − 0.523i)25-s − 0.192i·27-s − 0.550·29-s + (0.966 − 0.557i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ 0.823 - 0.567i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.262111946\)
\(L(\frac12)\) \(\approx\) \(1.262111946\)
\(L(4)\) 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 \( 1 + (-13.5 + 7.79i)T \)
7 \( 1 + (335. - 70.2i)T \)
good5 \( 1 + (68.1 + 39.3i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (457. + 791. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 87.6iT - 4.82e6T^{2} \)
17 \( 1 + (6.25e3 - 3.61e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-3.88e3 - 2.24e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-3.46e3 + 6.00e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 1.34e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.87e4 + 1.66e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.41e4 - 5.92e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 7.30e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.37e4T + 6.32e9T^{2} \)
47 \( 1 + (-9.39e4 - 5.42e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-7.40e4 - 1.28e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.12e5 + 1.22e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (8.32e4 + 4.80e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (3.22e4 + 5.58e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 5.41e4T + 1.28e11T^{2} \)
73 \( 1 + (-4.60e5 + 2.65e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (3.25e5 - 5.64e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 5.72e3iT - 3.26e11T^{2} \)
89 \( 1 + (-4.30e5 - 2.48e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 8.21e5iT - 8.32e11T^{2} \)
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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

−10.56914818760971211418495719944, −9.572577631248615053298014003742, −8.633301419202056795869375991374, −7.977024467522691096765690064825, −6.76911534090974628139634476802, −5.94200761250443535313948608324, −4.45557513112440823727432649363, −3.40689461310276343446424631652, −2.37363659432115446750852436546, −0.77291856939982006494036001325, 0.37599541572260257325003154814, 2.24624557512753211269994683361, 3.29007433898478648829891145914, 4.21389574580403756123017196531, 5.44172371540689967176313754983, 6.98567347551597366631250446329, 7.33792074427594901476918601074, 8.751298812012064373946269974280, 9.472367327952092476035289574585, 10.36212154972605662902685772468

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