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Properties

Label 2-336-21.17-c3-0-41
Degree $2$
Conductor $336$
Sign $-0.988 + 0.148i$
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  + (−2.82 − 4.36i)3-s + (2.24 + 3.88i)5-s + (9.71 − 15.7i)7-s + (−11.0 + 24.6i)9-s + (−20.2 − 11.7i)11-s − 5.91i·13-s + (10.6 − 20.7i)15-s + (58.0 − 100. i)17-s + (−8.02 + 4.63i)19-s + (−96.2 + 2.12i)21-s + (−107. + 62.1i)23-s + (52.4 − 90.7i)25-s + (138. − 21.1i)27-s + 207. i·29-s + (−122. − 70.8i)31-s + ⋯
L(s)  = 1  + (−0.543 − 0.839i)3-s + (0.200 + 0.347i)5-s + (0.524 − 0.851i)7-s + (−0.410 + 0.912i)9-s + (−0.555 − 0.320i)11-s − 0.126i·13-s + (0.183 − 0.357i)15-s + (0.828 − 1.43i)17-s + (−0.0969 + 0.0559i)19-s + (−0.999 + 0.0220i)21-s + (−0.976 + 0.563i)23-s + (0.419 − 0.726i)25-s + (0.988 − 0.150i)27-s + 1.33i·29-s + (−0.711 − 0.410i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.148i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.988 + 0.148i$
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: \((2,\ 336,\ (\ :3/2),\ -0.988 + 0.148i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7973508622\)
\(L(\frac12)\) \(\approx\) \(0.7973508622\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.82 + 4.36i)T \)
7 \( 1 + (-9.71 + 15.7i)T \)
good5 \( 1 + (-2.24 - 3.88i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20.2 + 11.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.91iT - 2.19e3T^{2} \)
17 \( 1 + (-58.0 + 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.02 - 4.63i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (107. - 62.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 207. iT - 2.43e4T^{2} \)
31 \( 1 + (122. + 70.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (149. + 259. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 508.T + 6.89e4T^{2} \)
43 \( 1 + 391.T + 7.95e4T^{2} \)
47 \( 1 + (-40.2 - 69.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-258. - 149. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (102. - 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (543. - 313. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (51.3 - 89.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 46.9iT - 3.57e5T^{2} \)
73 \( 1 + (228. + 131. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (533. + 924. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 270.T + 5.71e5T^{2} \)
89 \( 1 + (-443. - 768. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 219. iT - 9.12e5T^{2} \)
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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

−10.72124341482613348658033123906, −10.09105185372787106595870056804, −8.574874583000945598833355857560, −7.53851094876469344889951467994, −7.02790157882808567600328550475, −5.74575445553275777396634971907, −4.88628219434330803436775262333, −3.19533775425971030335348089066, −1.70315275104113337529818475851, −0.29728830942985014234435861566, 1.78116400758471389278484223006, 3.47001823227178497090753691653, 4.77436332532581650779458742573, 5.50182838370166174421651450706, 6.43555565283850149230140318399, 8.086148466145768190457377582444, 8.765341213041528771290091017192, 9.921914617231357085749304007919, 10.47321686943974487208152236406, 11.64497272804286116131335532790

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