Properties

Label 2-336-7.2-c3-0-17
Degree $2$
Conductor $336$
Sign $0.966 + 0.254i$
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  + (1.5 + 2.59i)3-s + (−0.363 + 0.629i)5-s + (18.1 − 3.72i)7-s + (−4.5 + 7.79i)9-s + (−32.2 − 55.8i)11-s + 71.8·13-s − 2.17·15-s + (−24.4 − 42.3i)17-s + (17.1 − 29.7i)19-s + (36.8 + 41.5i)21-s + (−0.451 + 0.782i)23-s + (62.2 + 107. i)25-s − 27·27-s + 226.·29-s + (−137. − 238. i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.0324 + 0.0562i)5-s + (0.979 − 0.201i)7-s + (−0.166 + 0.288i)9-s + (−0.883 − 1.53i)11-s + 1.53·13-s − 0.0375·15-s + (−0.348 − 0.604i)17-s + (0.207 − 0.359i)19-s + (0.383 + 0.431i)21-s + (−0.00409 + 0.00709i)23-s + (0.497 + 0.862i)25-s − 0.192·27-s + 1.45·29-s + (−0.798 − 1.38i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.966 + 0.254i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.966 + 0.254i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.274150623\)
\(L(\frac12)\) \(\approx\) \(2.274150623\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (-18.1 + 3.72i)T \)
good5 \( 1 + (0.363 - 0.629i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (32.2 + 55.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 71.8T + 2.19e3T^{2} \)
17 \( 1 + (24.4 + 42.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-17.1 + 29.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (0.451 - 0.782i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + (137. + 238. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (147. - 255. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 - 455.T + 7.95e4T^{2} \)
47 \( 1 + (-141. + 244. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (178. + 308. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-364. - 631. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (137. - 237. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-96.6 - 167. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 40.5T + 3.57e5T^{2} \)
73 \( 1 + (-103. - 178. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-468. + 812. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 911.T + 5.71e5T^{2} \)
89 \( 1 + (-474. + 822. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 39.4T + 9.12e5T^{2} \)
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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

−11.06362423114967533470733356326, −10.41252966152555536069980956296, −8.944776432400480328664132112879, −8.453677443578774643536403966511, −7.48571753886971300450265396039, −6.01431554544692317482270444283, −5.09668361546196792047619517266, −3.87659473187673759070614788282, −2.74042816234852921617269783653, −0.921609292584952222793063866179, 1.32809455635819595112327392463, 2.43217726454595964868180081977, 4.08004250355993672813326980244, 5.17356050305491726130382874634, 6.39024607845131903317091061151, 7.50529324764884074025547508214, 8.272879141020527939500917282237, 9.062943738327908120902860975589, 10.46273647816568575726665601582, 11.01873589237617853207120718408

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