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Properties

Label 2-336-7.2-c3-0-11
Degree $2$
Conductor $336$
Sign $0.999 - 0.0316i$
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 + (−4.91 + 8.50i)5-s + (16.3 − 8.74i)7-s + (−4.5 + 7.79i)9-s + (−7.08 − 12.2i)11-s − 26.1·13-s + 29.4·15-s + (39.2 + 68.0i)17-s + (36.5 − 63.3i)19-s + (−47.2 − 29.2i)21-s + (−48 + 83.1i)23-s + (14.2 + 24.6i)25-s + 27·27-s + 173.·29-s + (33.6 + 58.2i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.439 + 0.761i)5-s + (0.881 − 0.472i)7-s + (−0.166 + 0.288i)9-s + (−0.194 − 0.336i)11-s − 0.557·13-s + 0.507·15-s + (0.560 + 0.971i)17-s + (0.441 − 0.765i)19-s + (−0.490 − 0.304i)21-s + (−0.435 + 0.753i)23-s + (0.113 + 0.197i)25-s + 0.192·27-s + 1.10·29-s + (0.194 + 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0316i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0316i$
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: \((2,\ 336,\ (\ :3/2),\ 0.999 - 0.0316i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.637285014\)
\(L(\frac12)\) \(\approx\) \(1.637285014\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.3 + 8.74i)T \)
good5 \( 1 + (4.91 - 8.50i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.08 + 12.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 26.1T + 2.19e3T^{2} \)
17 \( 1 + (-39.2 - 68.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-36.5 + 63.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (48 - 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 + (-33.6 - 58.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 472.T + 6.89e4T^{2} \)
43 \( 1 - 463.T + 7.95e4T^{2} \)
47 \( 1 + (45.5 - 78.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-81.6 - 141. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (300. + 520. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (285. - 495. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (269. + 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (-221. - 383. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (22.8 - 39.5i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 686.T + 5.71e5T^{2} \)
89 \( 1 + (330. - 571. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 658.T + 9.12e5T^{2} \)
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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

−11.01852608161541470964292820629, −10.61908258908429942715881758253, −9.256814671198012256959424798619, −7.83276136537124684542835303103, −7.55553637373397120580651562594, −6.36420146142533051544538610071, −5.23029389241646082334502321492, −3.96421730892818759316881990939, −2.57059613881110536750646480577, −0.977613465305263028358899522633, 0.847399117709408841325485913908, 2.62454143272499928125932940399, 4.36027470163902780179713220034, 4.92820625694919743094041158263, 5.99424193187340429351892347169, 7.55882750348679987614215356424, 8.270023108696659880236441288322, 9.299043818855692869444618077997, 10.14992080655513201040228183804, 11.22689122056424992399044563843

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