Properties

Label 2-304-304.197-c1-0-31
Degree $2$
Conductor $304$
Sign $0.704 + 0.709i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.836 − 1.14i)2-s + (2.79 + 0.748i)3-s + (−0.601 − 1.90i)4-s + (0.209 − 0.781i)5-s + (3.18 − 2.55i)6-s + 3.41i·7-s + (−2.67 − 0.908i)8-s + (4.64 + 2.68i)9-s + (−0.715 − 0.891i)10-s + (−2.46 − 2.46i)11-s + (−0.253 − 5.77i)12-s + (1.10 + 4.12i)13-s + (3.89 + 2.85i)14-s + (1.16 − 2.02i)15-s + (−3.27 + 2.29i)16-s + (−3.56 − 6.17i)17-s + ⋯
L(s)  = 1  + (0.591 − 0.806i)2-s + (1.61 + 0.432i)3-s + (−0.300 − 0.953i)4-s + (0.0935 − 0.349i)5-s + (1.30 − 1.04i)6-s + 1.29i·7-s + (−0.947 − 0.321i)8-s + (1.54 + 0.893i)9-s + (−0.226 − 0.281i)10-s + (−0.743 − 0.743i)11-s + (−0.0732 − 1.66i)12-s + (0.306 + 1.14i)13-s + (1.04 + 0.763i)14-s + (0.301 − 0.522i)15-s + (−0.818 + 0.574i)16-s + (−0.865 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36908 - 0.986308i\)
\(L(\frac12)\) \(\approx\) \(2.36908 - 0.986308i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.836 + 1.14i)T \)
19 \( 1 + (4.19 + 1.16i)T \)
good3 \( 1 + (-2.79 - 0.748i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.209 + 0.781i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 + (2.46 + 2.46i)T + 11iT^{2} \)
13 \( 1 + (-1.10 - 4.12i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.56 + 6.17i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.33 + 3.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.994 - 3.70i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + (0.614 + 0.614i)T + 37iT^{2} \)
41 \( 1 + (-1.56 + 0.903i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.03 + 3.86i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.06 - 7.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.33 + 1.96i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.589 - 2.19i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.69 + 6.31i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.87 - 14.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.75 + 1.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.3 + 6.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.70 + 2.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.36 - 5.36i)T - 83iT^{2} \)
89 \( 1 + (-4.94 - 2.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.45 - 7.71i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.68188451547641322091469847914, −10.67345948507303232292597898010, −9.485343941782372433044916903889, −8.919491553709283721972162170317, −8.391508813857609046990874372766, −6.56740944305233402221026190607, −5.14748906587908679697778431321, −4.19789945682998529750691081237, −2.81574040469837555795462988875, −2.22968311621244569492171672347, 2.32349253796811652853640157126, 3.59154752294459853673621271564, 4.39251531392399284149885241279, 6.22948564162001652208838673385, 7.14934346688542491226676056367, 8.087393691263734057782716104317, 8.332328096668688584954330510555, 9.897413270951341033300566204003, 10.64908982940464882974700236598, 12.47603682044303684269109030666

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