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Properties

Label 2-336-21.17-c3-0-10
Degree $2$
Conductor $336$
Sign $0.225 - 0.974i$
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  + (−3.29 − 4.01i)3-s + (8.29 + 14.3i)5-s + (−9.36 − 15.9i)7-s + (−5.29 + 26.4i)9-s + (36.0 + 20.8i)11-s − 71.9i·13-s + (30.4 − 80.6i)15-s + (−46.0 + 79.8i)17-s + (−86.6 + 50.0i)19-s + (−33.3 + 90.2i)21-s + (−74.8 + 43.2i)23-s + (−75.1 + 130. i)25-s + (123. − 65.9i)27-s + 45.0i·29-s + (110. + 63.8i)31-s + ⋯
L(s)  = 1  + (−0.634 − 0.773i)3-s + (0.742 + 1.28i)5-s + (−0.505 − 0.862i)7-s + (−0.196 + 0.980i)9-s + (0.987 + 0.570i)11-s − 1.53i·13-s + (0.523 − 1.38i)15-s + (−0.657 + 1.13i)17-s + (−1.04 + 0.603i)19-s + (−0.346 + 0.938i)21-s + (−0.678 + 0.391i)23-s + (−0.601 + 1.04i)25-s + (0.882 − 0.470i)27-s + 0.288i·29-s + (0.641 + 0.370i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.225 - 0.974i$
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: \((2,\ 336,\ (\ :3/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.149568998\)
\(L(\frac12)\) \(\approx\) \(1.149568998\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.29 + 4.01i)T \)
7 \( 1 + (9.36 + 15.9i)T \)
good5 \( 1 + (-8.29 - 14.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.0 - 20.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 71.9iT - 2.19e3T^{2} \)
17 \( 1 + (46.0 - 79.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (86.6 - 50.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (74.8 - 43.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 45.0iT - 2.43e4T^{2} \)
31 \( 1 + (-110. - 63.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-188. - 326. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 + (-103. - 178. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-191. - 110. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (303. - 525. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (291. - 168. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (15.4 - 26.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 450. iT - 3.57e5T^{2} \)
73 \( 1 + (-613. - 354. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-372. - 645. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + (333. + 577. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.16e3iT - 9.12e5T^{2} \)
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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

−11.10334437176649946059938450808, −10.43502162169065918507257166264, −9.936363244981499602465816860665, −8.238980358253140842333258765528, −7.23594192012912060838084063853, −6.39012416902577157427185372381, −5.96518210266927314866829567189, −4.17094532994111095348224397043, −2.74766949437557412935608349385, −1.38604850939327461827039834442, 0.45799813614825903802359177318, 2.19252389811793699022251567236, 4.10618911653414880710608979587, 4.86293339741555581454108056388, 6.02265010155574250396441420604, 6.55942528426864244396089620371, 8.689873510555724711603972704966, 9.235267828436538336136699645503, 9.576725442155139734513376924702, 11.09182456664987889554736307093

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