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Properties

Label 2-336-21.5-c3-0-0
Degree $2$
Conductor $336$
Sign $-0.964 - 0.264i$
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  + (−4.30 − 2.90i)3-s + (−7.41 + 12.8i)5-s + (16.6 − 8.10i)7-s + (10.1 + 25.0i)9-s + (−0.589 + 0.340i)11-s − 52.4i·13-s + (69.2 − 33.7i)15-s + (51.6 + 89.5i)17-s + (−86.7 − 50.0i)19-s + (−95.2 − 13.4i)21-s + (3.63 + 2.10i)23-s + (−47.4 − 82.2i)25-s + (29.2 − 137. i)27-s − 53.5i·29-s + (−235. + 136. i)31-s + ⋯
L(s)  = 1  + (−0.828 − 0.559i)3-s + (−0.663 + 1.14i)5-s + (0.899 − 0.437i)7-s + (0.374 + 0.927i)9-s + (−0.0161 + 0.00932i)11-s − 1.11i·13-s + (1.19 − 0.581i)15-s + (0.737 + 1.27i)17-s + (−1.04 − 0.604i)19-s + (−0.990 − 0.140i)21-s + (0.0329 + 0.0190i)23-s + (−0.379 − 0.657i)25-s + (0.208 − 0.978i)27-s − 0.342i·29-s + (−1.36 + 0.789i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.964 - 0.264i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.964 - 0.264i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1657607528\)
\(L(\frac12)\) \(\approx\) \(0.1657607528\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.30 + 2.90i)T \)
7 \( 1 + (-16.6 + 8.10i)T \)
good5 \( 1 + (7.41 - 12.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (0.589 - 0.340i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 52.4iT - 2.19e3T^{2} \)
17 \( 1 + (-51.6 - 89.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (86.7 + 50.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-3.63 - 2.10i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 53.5iT - 2.43e4T^{2} \)
31 \( 1 + (235. - 136. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-25.2 + 43.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 318.T + 6.89e4T^{2} \)
43 \( 1 + 168.T + 7.95e4T^{2} \)
47 \( 1 + (313. - 543. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (625. - 361. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (263. + 455. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-205. - 118. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-149. - 258. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 379. iT - 3.57e5T^{2} \)
73 \( 1 + (559. - 323. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-211. + 366. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 391.T + 5.71e5T^{2} \)
89 \( 1 + (-316. + 548. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.43e3iT - 9.12e5T^{2} \)
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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

−11.24859289154710528751094946056, −10.86749053054283332603262359339, −10.21633095904310288688660828225, −8.284100684523651121659384821035, −7.67878043351305744307016770650, −6.81812926368634490290093965445, −5.81339067640765565298985335780, −4.60829821845339985920262672602, −3.27322842652311513757966153395, −1.61632570407636255572883128080, 0.06777483802828525081241196429, 1.59555367792608793782634159577, 3.83381838415034255205461500860, 4.79241481435841184394326831384, 5.36520909999183377363785258172, 6.74809846486800033294550696894, 8.012524592085113217805282637643, 8.888719614578863171273808048205, 9.685441219533546359729085654080, 10.93709751479674431904180842755

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