Properties

Label 2-336-21.17-c3-0-28
Degree $2$
Conductor $336$
Sign $0.0952 + 0.995i$
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  + (−3.24 + 4.05i)3-s + (3.20 + 5.55i)5-s + (−14.8 + 11.1i)7-s + (−5.89 − 26.3i)9-s + (−16.2 − 9.39i)11-s − 3.82i·13-s + (−32.9 − 5.03i)15-s + (−8.12 + 14.0i)17-s + (−129. + 74.9i)19-s + (3.01 − 96.1i)21-s + (109. − 63.3i)23-s + (41.9 − 72.6i)25-s + (126. + 61.6i)27-s − 168. i·29-s + (43.4 + 25.0i)31-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)3-s + (0.286 + 0.496i)5-s + (−0.799 + 0.600i)7-s + (−0.218 − 0.975i)9-s + (−0.446 − 0.257i)11-s − 0.0816i·13-s + (−0.567 − 0.0867i)15-s + (−0.115 + 0.200i)17-s + (−1.56 + 0.904i)19-s + (0.0313 − 0.999i)21-s + (0.994 − 0.573i)23-s + (0.335 − 0.581i)25-s + (0.898 + 0.439i)27-s − 1.07i·29-s + (0.251 + 0.145i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0952 + 0.995i$
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: \((2,\ 336,\ (\ :3/2),\ 0.0952 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3373914868\)
\(L(\frac12)\) \(\approx\) \(0.3373914868\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.24 - 4.05i)T \)
7 \( 1 + (14.8 - 11.1i)T \)
good5 \( 1 + (-3.20 - 5.55i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (16.2 + 9.39i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 3.82iT - 2.19e3T^{2} \)
17 \( 1 + (8.12 - 14.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (129. - 74.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-109. + 63.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 168. iT - 2.43e4T^{2} \)
31 \( 1 + (-43.4 - 25.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-24.7 - 42.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 19.9T + 6.89e4T^{2} \)
43 \( 1 + 5.14T + 7.95e4T^{2} \)
47 \( 1 + (308. + 534. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-242. - 140. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-274. + 475. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (507. - 293. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-378. + 655. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 351. iT - 3.57e5T^{2} \)
73 \( 1 + (530. + 306. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (505. + 876. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.01e3T + 5.71e5T^{2} \)
89 \( 1 + (-178. - 309. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 228. iT - 9.12e5T^{2} \)
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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

−10.58369470719917559619660463041, −10.28317011752287469544318781950, −9.186990216220456191738037803258, −8.288526375766430971702824060586, −6.56765433542586443269251285640, −6.13877700820626168601299957850, −4.96389900504380557239522454002, −3.69107057820647225063730807252, −2.49986552914975391124217365810, −0.13868954231622725648851092688, 1.22328633290270394982432762442, 2.75320744227479287227999478792, 4.49663205027693978435847478337, 5.49885803281426991155879322043, 6.66511282023917731036665299007, 7.23403912254538820515880065793, 8.508168284799488446612173712513, 9.488452471157976213166762113523, 10.62798575942207389798989977025, 11.24044227779655621388994145350

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