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Properties

Label 2-336-7.5-c6-0-41
Degree $2$
Conductor $336$
Sign $-0.493 + 0.869i$
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 + (63.2 + 36.5i)5-s + (271. − 209. i)7-s + (121.5 − 210. i)9-s + (−1.07e3 − 1.85e3i)11-s + 2.55e3i·13-s + 1.13e3·15-s + (2.87e3 − 1.65e3i)17-s + (−3.81e3 − 2.20e3i)19-s + (2.02e3 − 4.94e3i)21-s + (8.54e3 − 1.48e4i)23-s + (−5.14e3 − 8.90e3i)25-s − 3.78e3i·27-s − 3.32e4·29-s + (−1.01e4 + 5.85e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.506 + 0.292i)5-s + (0.791 − 0.611i)7-s + (0.166 − 0.288i)9-s + (−0.806 − 1.39i)11-s + 1.16i·13-s + 0.337·15-s + (0.584 − 0.337i)17-s + (−0.555 − 0.320i)19-s + (0.218 − 0.534i)21-s + (0.702 − 1.21i)23-s + (−0.329 − 0.570i)25-s − 0.192i·27-s − 1.36·29-s + (−0.340 + 0.196i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.493 + 0.869i$
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: \((2,\ 336,\ (\ :3),\ -0.493 + 0.869i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.267410796\)
\(L(\frac12)\) \(\approx\) \(2.267410796\)
\(L(4)\) 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 \( 1 + (-271. + 209. i)T \)
good5 \( 1 + (-63.2 - 36.5i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (1.07e3 + 1.85e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 2.55e3iT - 4.82e6T^{2} \)
17 \( 1 + (-2.87e3 + 1.65e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (3.81e3 + 2.20e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-8.54e3 + 1.48e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 3.32e4T + 5.94e8T^{2} \)
31 \( 1 + (1.01e4 - 5.85e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (2.43e4 - 4.21e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 7.14e4iT - 4.75e9T^{2} \)
43 \( 1 + 8.68e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.43e5 - 8.27e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-4.24e4 - 7.34e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.85e4 + 1.07e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.35e5 - 1.35e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.39e5 + 4.14e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 3.37e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.40e5 + 1.96e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-5.03e3 + 8.72e3i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 7.77e4iT - 3.26e11T^{2} \)
89 \( 1 + (-4.78e5 - 2.76e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.06e6iT - 8.32e11T^{2} \)
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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

−10.40130130474866253915025509495, −9.110138587836049341811042419314, −8.379886847824015449221724871349, −7.40721259188766215432106142442, −6.46369489441112695622139900080, −5.30804092589813264565664288042, −4.08157100236239789325979567223, −2.82104423943087157617728077028, −1.75526113336205906860092649121, −0.45058623985018949228873426485, 1.53404525525762185280715726229, 2.37059064419993134514859202618, 3.74147930243281583037805518402, 5.18392051809889783057483934246, 5.52740671570931141042449888930, 7.36885879796938531793126016727, 7.987204888141564105815675937692, 9.044752505609986254426448266372, 9.872475594494099271358027820480, 10.63206516715028859976265956128

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