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Properties

Label 2-300-4.3-c4-0-26
Degree $2$
Conductor $300$
Sign $0.999 - 0.0161i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.80 − 2.85i)2-s − 5.19i·3-s + (−0.257 − 15.9i)4-s + (−14.8 − 14.5i)6-s + 63.6i·7-s + (−46.3 − 44.1i)8-s − 27·9-s + 220. i·11-s + (−83.1 + 1.33i)12-s + 236.·13-s + (181. + 178. i)14-s + (−255. + 8.24i)16-s − 46.7·17-s + (−75.7 + 76.9i)18-s + 195. i·19-s + ⋯
L(s)  = 1  + (0.701 − 0.712i)2-s − 0.577i·3-s + (−0.0161 − 0.999i)4-s + (−0.411 − 0.404i)6-s + 1.29i·7-s + (−0.723 − 0.689i)8-s − 0.333·9-s + 1.82i·11-s + (−0.577 + 0.00929i)12-s + 1.40·13-s + (0.925 + 0.910i)14-s + (−0.999 + 0.0322i)16-s − 0.161·17-s + (−0.233 + 0.237i)18-s + 0.542i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0161i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 - 0.0161i$
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: \((2,\ 300,\ (\ :2),\ 0.999 - 0.0161i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.505184400\)
\(L(\frac12)\) \(\approx\) \(2.505184400\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.80 + 2.85i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 63.6iT - 2.40e3T^{2} \)
11 \( 1 - 220. iT - 1.46e4T^{2} \)
13 \( 1 - 236.T + 2.85e4T^{2} \)
17 \( 1 + 46.7T + 8.35e4T^{2} \)
19 \( 1 - 195. iT - 1.30e5T^{2} \)
23 \( 1 - 741. iT - 2.79e5T^{2} \)
29 \( 1 - 1.05e3T + 7.07e5T^{2} \)
31 \( 1 + 1.06e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + 1.14e3T + 2.82e6T^{2} \)
43 \( 1 - 1.24e3iT - 3.41e6T^{2} \)
47 \( 1 + 406. iT - 4.87e6T^{2} \)
53 \( 1 - 1.44e3T + 7.89e6T^{2} \)
59 \( 1 + 2.47e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.81e3T + 1.38e7T^{2} \)
67 \( 1 - 5.79e3iT - 2.01e7T^{2} \)
71 \( 1 - 309. iT - 2.54e7T^{2} \)
73 \( 1 - 996.T + 2.83e7T^{2} \)
79 \( 1 - 1.09e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.09e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.09e4T + 6.27e7T^{2} \)
97 \( 1 + 1.23e4T + 8.85e7T^{2} \)
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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

−11.53095644194898756662611622536, −10.25648841631251630754689975168, −9.372568395095941742713613351063, −8.373302544780349774854115563713, −6.94632965985665888782684754782, −5.96647516660060957480890849811, −5.06935126900720321765459194618, −3.70759195152395603253693169139, −2.32638869113890903763332798644, −1.47959235140008135510407308550, 0.62960682395157647320103507667, 3.17822395214108824175382697006, 3.87715095715542620655572111602, 4.98809886499090415599418671091, 6.18121698945791866786761456623, 6.91992314797824692539352447829, 8.454461328346236197122141403336, 8.662658933981868206755470811273, 10.54140087114867818711421099092, 10.92518323946201223493350218325

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