↑

Properties

Label 2-336-21.5-c3-0-2
Degree $2$
Conductor $336$
Sign $-0.772 - 0.634i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.72 + 3.61i)3-s + (6.11 − 10.5i)5-s + (−4.39 − 17.9i)7-s + (0.811 − 26.9i)9-s + (−47.3 + 27.3i)11-s + 27.0i·13-s + (15.5 + 61.5i)15-s + (−20.3 − 35.1i)17-s + (48.4 + 27.9i)19-s + (81.4 + 51.2i)21-s + (93.3 + 53.8i)23-s + (−12.1 − 21.1i)25-s + (94.6 + 103. i)27-s + 38.2i·29-s + (−257. + 148. i)31-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)3-s + (0.546 − 0.946i)5-s + (−0.237 − 0.971i)7-s + (0.0300 − 0.999i)9-s + (−1.29 + 0.749i)11-s + 0.576i·13-s + (0.267 + 1.06i)15-s + (−0.289 − 0.501i)17-s + (0.585 + 0.338i)19-s + (0.846 + 0.532i)21-s + (0.846 + 0.488i)23-s + (−0.0975 − 0.168i)25-s + (0.674 + 0.738i)27-s + 0.244i·29-s + (−1.49 + 0.862i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.772 - 0.634i$
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: \((2,\ 336,\ (\ :3/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3894694529\)
\(L(\frac12)\) \(\approx\) \(0.3894694529\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.72 - 3.61i)T \)
7 \( 1 + (4.39 + 17.9i)T \)
good5 \( 1 + (-6.11 + 10.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (47.3 - 27.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 27.0iT - 2.19e3T^{2} \)
17 \( 1 + (20.3 + 35.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.4 - 27.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-93.3 - 53.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 38.2iT - 2.43e4T^{2} \)
31 \( 1 + (257. - 148. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (142. - 246. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 28.3T + 6.89e4T^{2} \)
43 \( 1 + 212.T + 7.95e4T^{2} \)
47 \( 1 + (125. - 216. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (294. - 170. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (451. + 781. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-499. - 288. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-184. - 319. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.18e3iT - 3.57e5T^{2} \)
73 \( 1 + (407. - 235. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-20.0 + 34.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 855.T + 5.71e5T^{2} \)
89 \( 1 + (322. - 557. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 748. iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32225996088561033792531377343, −10.48064653980681781689147568596, −9.708806501829994272158013231818, −9.034873762445381583425286473344, −7.57717237889727943623326906334, −6.60887248699017872162353061363, −5.12463851151527427766724683689, −4.89111559376849136584016034624, −3.41951246024243276788717364431, −1.38964201969145314719771250563, 0.15200471664369921111934360870, 2.16012617174956775507226509107, 3.04640761069816086286338217291, 5.26362839565827770618310978614, 5.80716019824085283543549071621, 6.75605735830887987160720421636, 7.75896732861251675519348325779, 8.794474060773086758314604284546, 10.14664023863437221883222115963, 10.84574888224752795454180458917

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