↑

Properties

Label 2-300-4.3-c4-0-48
Degree $2$
Conductor $300$
Sign $0.915 - 0.401i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
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  + (3.34 + 2.18i)2-s − 5.19i·3-s + (6.43 + 14.6i)4-s + (11.3 − 17.4i)6-s + 1.39i·7-s + (−10.5 + 63.1i)8-s − 27·9-s − 164. i·11-s + (76.1 − 33.4i)12-s + 158.·13-s + (−3.05 + 4.67i)14-s + (−173. + 188. i)16-s + 548.·17-s + (−90.4 − 59.0i)18-s − 25.6i·19-s + ⋯
L(s)  = 1  + (0.837 + 0.546i)2-s − 0.577i·3-s + (0.401 + 0.915i)4-s + (0.315 − 0.483i)6-s + 0.0284i·7-s + (−0.164 + 0.986i)8-s − 0.333·9-s − 1.36i·11-s + (0.528 − 0.232i)12-s + 0.938·13-s + (−0.0155 + 0.0238i)14-s + (−0.676 + 0.736i)16-s + 1.89·17-s + (−0.279 − 0.182i)18-s − 0.0711i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.915 - 0.401i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.546565430\)
\(L(\frac12)\) \(\approx\) \(3.546565430\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.34 - 2.18i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 1.39iT - 2.40e3T^{2} \)
11 \( 1 + 164. iT - 1.46e4T^{2} \)
13 \( 1 - 158.T + 2.85e4T^{2} \)
17 \( 1 - 548.T + 8.35e4T^{2} \)
19 \( 1 + 25.6iT - 1.30e5T^{2} \)
23 \( 1 - 730. iT - 2.79e5T^{2} \)
29 \( 1 - 773.T + 7.07e5T^{2} \)
31 \( 1 + 194. iT - 9.23e5T^{2} \)
37 \( 1 - 1.37e3T + 1.87e6T^{2} \)
41 \( 1 + 647.T + 2.82e6T^{2} \)
43 \( 1 + 1.50e3iT - 3.41e6T^{2} \)
47 \( 1 + 647. iT - 4.87e6T^{2} \)
53 \( 1 - 4.40e3T + 7.89e6T^{2} \)
59 \( 1 - 5.97e3iT - 1.21e7T^{2} \)
61 \( 1 + 560.T + 1.38e7T^{2} \)
67 \( 1 - 2.98e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.67e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.64e3T + 2.83e7T^{2} \)
79 \( 1 + 3.15e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.30e4iT - 4.74e7T^{2} \)
89 \( 1 + 7.11e3T + 6.27e7T^{2} \)
97 \( 1 + 5.31e3T + 8.85e7T^{2} \)
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

−11.59096103404660229263803544732, −10.45812575166350755271129090742, −8.869175232898134194285310888499, −8.070598020964429243994715362751, −7.22083872435305188349440649145, −5.94565223803027208199687426704, −5.55537515935006795797737073084, −3.81251339212111181320367205512, −2.94229018099007356172275947034, −1.11219033850203479403140953174, 1.11938681543897745432313218844, 2.64477781017275777970719183618, 3.84451331128542449359083331965, 4.75454423195855292733306033861, 5.78741914356767626730609234793, 6.87639433648814828264861116102, 8.223589226553319146683596973433, 9.620735801087909697662129120305, 10.17873530608519783686242630708, 11.03170518752591758231634258114

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