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Properties

Label 2-336-21.17-c3-0-38
Degree $2$
Conductor $336$
Sign $-0.0770 + 0.997i$
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  + (4.97 − 1.49i)3-s + (−0.619 − 1.07i)5-s + (−8.82 + 16.2i)7-s + (22.5 − 14.9i)9-s + (−56.1 − 32.4i)11-s − 54.9i·13-s + (−4.69 − 4.41i)15-s + (47.9 − 82.9i)17-s + (−23.2 + 13.4i)19-s + (−19.4 + 94.2i)21-s + (150. − 86.6i)23-s + (61.7 − 106. i)25-s + (89.6 − 107. i)27-s − 16.9i·29-s + (−66.1 − 38.1i)31-s + ⋯
L(s)  = 1  + (0.957 − 0.288i)3-s + (−0.0554 − 0.0959i)5-s + (−0.476 + 0.879i)7-s + (0.833 − 0.552i)9-s + (−1.53 − 0.888i)11-s − 1.17i·13-s + (−0.0807 − 0.0759i)15-s + (0.683 − 1.18i)17-s + (−0.281 + 0.162i)19-s + (−0.202 + 0.979i)21-s + (1.36 − 0.785i)23-s + (0.493 − 0.855i)25-s + (0.638 − 0.769i)27-s − 0.108i·29-s + (−0.383 − 0.221i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0770 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0770 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.0770 + 0.997i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.0770 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.942745800\)
\(L(\frac12)\) \(\approx\) \(1.942745800\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.97 + 1.49i)T \)
7 \( 1 + (8.82 - 16.2i)T \)
good5 \( 1 + (0.619 + 1.07i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (56.1 + 32.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 54.9iT - 2.19e3T^{2} \)
17 \( 1 + (-47.9 + 82.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23.2 - 13.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-150. + 86.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 16.9iT - 2.43e4T^{2} \)
31 \( 1 + (66.1 + 38.1i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-69.3 - 120. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 100.T + 6.89e4T^{2} \)
43 \( 1 + 197.T + 7.95e4T^{2} \)
47 \( 1 + (-161. - 280. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-248. - 143. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-84.2 + 145. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (166. - 96.1i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (442. - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 661. iT - 3.57e5T^{2} \)
73 \( 1 + (-125. - 72.2i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (228. + 395. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 967.T + 5.71e5T^{2} \)
89 \( 1 + (605. + 1.04e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 777. iT - 9.12e5T^{2} \)
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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

−10.69846324779681479345364843844, −9.895356252568164007402147038194, −8.787258751013922695227075748481, −8.204129399723900109320143913770, −7.26553273480599174587306975150, −5.93915738085901648565935313706, −4.94400273811267474048264252647, −3.02962311354107885460866945459, −2.73533972315050766053938166798, −0.60472237426237063462739737041, 1.70515644110826213685323374163, 3.09352736597239194338829969736, 4.11519894910420539082161355766, 5.20802113180185061405548808522, 6.97577062158918917811006285128, 7.47126948216938048221669506320, 8.587912731790632977019895843522, 9.594560756585177885428982773468, 10.31317405797908233627686896516, 11.04946512561731049480816123286

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