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Properties

Label 2-336-12.11-c3-0-30
Degree $2$
Conductor $336$
Sign $-0.297 + 0.954i$
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  + (1.13 + 5.06i)3-s − 14.6i·5-s + 7i·7-s + (−24.4 + 11.5i)9-s + 13.9·11-s − 45.4·13-s + (74.2 − 16.6i)15-s − 120. i·17-s − 34.9i·19-s + (−35.4 + 7.97i)21-s − 151.·23-s − 89.4·25-s + (−86.3 − 110. i)27-s + 224. i·29-s − 44.2i·31-s + ⋯
L(s)  = 1  + (0.219 + 0.975i)3-s − 1.30i·5-s + 0.377i·7-s + (−0.903 + 0.427i)9-s + 0.381·11-s − 0.970·13-s + (1.27 − 0.287i)15-s − 1.72i·17-s − 0.421i·19-s + (−0.368 + 0.0829i)21-s − 1.37·23-s − 0.715·25-s + (−0.615 − 0.787i)27-s + 1.43i·29-s − 0.256i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.297 + 0.954i$
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: \((2,\ 336,\ (\ :3/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8869202333\)
\(L(\frac12)\) \(\approx\) \(0.8869202333\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.13 - 5.06i)T \)
7 \( 1 - 7iT \)
good5 \( 1 + 14.6iT - 125T^{2} \)
11 \( 1 - 13.9T + 1.33e3T^{2} \)
13 \( 1 + 45.4T + 2.19e3T^{2} \)
17 \( 1 + 120. iT - 4.91e3T^{2} \)
19 \( 1 + 34.9iT - 6.85e3T^{2} \)
23 \( 1 + 151.T + 1.21e4T^{2} \)
29 \( 1 - 224. iT - 2.43e4T^{2} \)
31 \( 1 + 44.2iT - 2.97e4T^{2} \)
37 \( 1 + 224.T + 5.06e4T^{2} \)
41 \( 1 + 459. iT - 6.89e4T^{2} \)
43 \( 1 + 497. iT - 7.95e4T^{2} \)
47 \( 1 - 134.T + 1.03e5T^{2} \)
53 \( 1 + 282. iT - 1.48e5T^{2} \)
59 \( 1 - 48.3T + 2.05e5T^{2} \)
61 \( 1 + 343.T + 2.26e5T^{2} \)
67 \( 1 + 678. iT - 3.00e5T^{2} \)
71 \( 1 - 820.T + 3.57e5T^{2} \)
73 \( 1 - 370.T + 3.89e5T^{2} \)
79 \( 1 - 986. iT - 4.93e5T^{2} \)
83 \( 1 + 484.T + 5.71e5T^{2} \)
89 \( 1 - 980. iT - 7.04e5T^{2} \)
97 \( 1 + 488.T + 9.12e5T^{2} \)
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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

−10.74989643595067485151895509112, −9.619481672226889297282124662238, −9.138517056264821923907492002716, −8.361243885959425187843071957632, −7.09813275014867596252290920237, −5.34988635768393002918865294318, −4.99293415423498818174125278261, −3.78351186823110116960914179723, −2.28387101692231050070707661822, −0.28976706962254746348352312432, 1.70628714633869976902055474491, 2.87024661512549964893876811850, 4.06210451778192390399491305875, 6.01672363458374175634160235737, 6.55783802090619345720864945283, 7.60791028328402807610479351771, 8.210566918188003581358698169622, 9.713539451262084619616054099223, 10.48899047405576493638244314056, 11.48539474731392090602834125894

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