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Properties

Label 2-336-84.23-c3-0-8
Degree $2$
Conductor $336$
Sign $0.987 - 0.156i$
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  + (−4.15 − 3.11i)3-s + (−17.2 − 9.94i)5-s + (−5.66 + 17.6i)7-s + (7.59 + 25.9i)9-s + (−22.0 − 38.2i)11-s − 32.6·13-s + (40.6 + 95.0i)15-s + (−78.8 + 45.5i)17-s + (−49.6 − 28.6i)19-s + (78.4 − 55.6i)21-s + (41.9 − 72.7i)23-s + (135. + 234. i)25-s + (49.1 − 131. i)27-s + 188. i·29-s + (225. − 130. i)31-s + ⋯
L(s)  = 1  + (−0.800 − 0.599i)3-s + (−1.54 − 0.889i)5-s + (−0.305 + 0.952i)7-s + (0.281 + 0.959i)9-s + (−0.604 − 1.04i)11-s − 0.696·13-s + (0.699 + 1.63i)15-s + (−1.12 + 0.649i)17-s + (−0.599 − 0.345i)19-s + (0.815 − 0.578i)21-s + (0.380 − 0.659i)23-s + (1.08 + 1.87i)25-s + (0.349 − 0.936i)27-s + 1.20i·29-s + (1.30 − 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.987 - 0.156i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4166524028\)
\(L(\frac12)\) \(\approx\) \(0.4166524028\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.15 + 3.11i)T \)
7 \( 1 + (5.66 - 17.6i)T \)
good5 \( 1 + (17.2 + 9.94i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (22.0 + 38.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 32.6T + 2.19e3T^{2} \)
17 \( 1 + (78.8 - 45.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (49.6 + 28.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-41.9 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 188. iT - 2.43e4T^{2} \)
31 \( 1 + (-225. + 130. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-172. + 299. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 34.4iT - 6.89e4T^{2} \)
43 \( 1 - 201. iT - 7.95e4T^{2} \)
47 \( 1 + (-232. + 402. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-176. + 102. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-113. - 197. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (423. - 732. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (146. - 84.6i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 86.4T + 3.57e5T^{2} \)
73 \( 1 + (112. + 194. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-513. - 296. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 127.T + 5.71e5T^{2} \)
89 \( 1 + (560. + 323. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 704.T + 9.12e5T^{2} \)
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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

−11.36364249131984036993523533196, −10.60284020206211721630846327503, −8.869070161775234030465858198076, −8.376274896227545925090872208063, −7.38254879014205700335227124984, −6.24779197618782006587368212081, −5.17798600973051161274141447221, −4.26310384243003586948632657658, −2.56915383824781916871731897442, −0.60894869948042683609322303515, 0.31061955574324728977652085182, 2.92878699239396238033754187403, 4.22033965462282921322645496858, 4.67771421184039805968483619464, 6.52654157281364379249598994888, 7.15661819038609589079195651539, 7.977434042270251312262143107604, 9.605030956937116876577283048083, 10.37535340446695426867406705241, 11.05202654540696441101762437705

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