↑

Properties

Label 2-304-19.12-c4-0-34
Degree $2$
Conductor $304$
Sign $-0.245 + 0.969i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (14.2 − 8.23i)3-s + (2.53 + 4.38i)5-s − 53.8·7-s + (95.2 − 164. i)9-s − 12.0·11-s + (200. + 115. i)13-s + (72.2 + 41.7i)15-s + (−170. − 296. i)17-s + (−131. − 336. i)19-s + (−768. + 443. i)21-s + (386. − 670. i)23-s + (299. − 519. i)25-s − 1.80e3i·27-s + (30.3 + 17.5i)29-s − 1.26e3i·31-s + ⋯
L(s)  = 1  + (1.58 − 0.915i)3-s + (0.101 + 0.175i)5-s − 1.09·7-s + (1.17 − 2.03i)9-s − 0.0992·11-s + (1.18 + 0.684i)13-s + (0.321 + 0.185i)15-s + (−0.591 − 1.02i)17-s + (−0.364 − 0.931i)19-s + (−1.74 + 1.00i)21-s + (0.731 − 1.26i)23-s + (0.479 − 0.830i)25-s − 2.47i·27-s + (0.0360 + 0.0208i)29-s − 1.31i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.245 + 0.969i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.993632429\)
\(L(\frac12)\) \(\approx\) \(2.993632429\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (131. + 336. i)T \)
good3 \( 1 + (-14.2 + 8.23i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-2.53 - 4.38i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 53.8T + 2.40e3T^{2} \)
11 \( 1 + 12.0T + 1.46e4T^{2} \)
13 \( 1 + (-200. - 115. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (170. + 296. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-386. + 670. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-30.3 - 17.5i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.26e3iT - 9.23e5T^{2} \)
37 \( 1 - 307. iT - 1.87e6T^{2} \)
41 \( 1 + (1.03e3 - 598. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (294. + 509. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.08e3 - 1.87e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.07e3 - 1.19e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.24e3 - 2.44e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.14e3 - 3.71e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.52e3 - 878. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-6.97e3 + 4.02e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-4.13e3 - 7.16e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.74e3 + 1.58e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 5.13e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.05e4 - 6.07e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-5.36e3 + 3.09e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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

−10.72341382649342358057301768831, −9.381058773040731305987445736802, −8.987849236724166904130698889053, −8.028875659116862651481805158527, −6.71644572632193795746645245663, −6.55252579791565209129505147173, −4.29556647087708937324563526922, −3.05858366692946790722890099921, −2.35088291408327268122741479662, −0.74273304031994909708169521617, 1.73751941226469679747019204193, 3.40350937685118539909656278644, 3.55218812661654470854135674291, 5.14621902783202372339532768829, 6.51391466582246881465385594437, 7.88625518832310039913719545672, 8.683945621491304439766740455135, 9.337031930598855593015190422355, 10.25037488703445805859970847693, 10.89608142243955615115748171391

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