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Properties

Label 2-336-7.6-c8-0-32
Degree $2$
Conductor $336$
Sign $0.280 + 0.959i$
Analytic cond. $136.879$
Root an. cond. $11.6995$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.7i·3-s − 316. i·5-s + (−2.30e3 + 674. i)7-s − 2.18e3·9-s − 1.02e4·11-s − 2.86e4i·13-s − 1.48e4·15-s + 1.04e5i·17-s + 2.17e5i·19-s + (3.15e4 + 1.07e5i)21-s + 1.82e5·23-s + 2.90e5·25-s + 1.02e5i·27-s − 2.64e5·29-s − 3.35e5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.507i·5-s + (−0.959 + 0.280i)7-s − 0.333·9-s − 0.702·11-s − 1.00i·13-s − 0.292·15-s + 1.25i·17-s + 1.67i·19-s + (0.162 + 0.554i)21-s + 0.651·23-s + 0.742·25-s + 0.192i·27-s − 0.374·29-s − 0.363i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(136.879\)
Root analytic conductor: \(11.6995\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :4),\ 0.280 + 0.959i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.287207022\)
\(L(\frac12)\) \(\approx\) \(1.287207022\)
\(L(5)\) 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 \( 1 + 46.7iT \)
7 \( 1 + (2.30e3 - 674. i)T \)
good5 \( 1 + 316. iT - 3.90e5T^{2} \)
11 \( 1 + 1.02e4T + 2.14e8T^{2} \)
13 \( 1 + 2.86e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.04e5iT - 6.97e9T^{2} \)
19 \( 1 - 2.17e5iT - 1.69e10T^{2} \)
23 \( 1 - 1.82e5T + 7.83e10T^{2} \)
29 \( 1 + 2.64e5T + 5.00e11T^{2} \)
31 \( 1 + 3.35e5iT - 8.52e11T^{2} \)
37 \( 1 + 3.14e6T + 3.51e12T^{2} \)
41 \( 1 + 1.29e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.44e6T + 1.16e13T^{2} \)
47 \( 1 - 2.53e6iT - 2.38e13T^{2} \)
53 \( 1 - 9.13e6T + 6.22e13T^{2} \)
59 \( 1 - 4.05e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.09e7iT - 1.91e14T^{2} \)
67 \( 1 - 1.07e7T + 4.06e14T^{2} \)
71 \( 1 + 2.50e7T + 6.45e14T^{2} \)
73 \( 1 - 1.50e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.95e7T + 1.51e15T^{2} \)
83 \( 1 - 3.29e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.02e8iT - 3.93e15T^{2} \)
97 \( 1 + 2.60e7iT - 7.83e15T^{2} \)
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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

−10.10771014856128823076236951059, −8.873109679252136430658410106734, −8.156946582734046886720317496543, −7.18817644635476032424478556501, −5.97657455801158766898413106583, −5.42634552313293936691604526481, −3.82644246555900714527298966994, −2.82486410477230285296207749043, −1.57366729040903317369656914898, −0.40620076624473904571087821169, 0.63139371903347768671417099683, 2.51427583479287744576937120474, 3.20139760505179886657583132126, 4.45873653269575246589675165174, 5.38767837581911179421223009832, 6.82651409475535811519739826246, 7.15925911048868164458332047147, 8.857722022256555220866226601106, 9.388128378427908250832181334964, 10.41327445785855110964724260722

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