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Properties

Label 2-300-15.2-c3-0-8
Degree $2$
Conductor $300$
Sign $0.572 - 0.820i$
Analytic cond. $17.7005$
Root an. cond. $4.20720$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.18 + 0.289i)3-s + (4.89 + 4.89i)7-s + (26.8 + 3.00i)9-s + 53.6i·11-s + (−39.1 + 39.1i)13-s + (32.8 − 32.8i)17-s − 28i·19-s + (24 + 26.8i)21-s + (115. + 115. i)23-s + (138. + 23.3i)27-s + 53.6·29-s − 56·31-s + (−15.5 + 278. i)33-s + (137. + 137. i)37-s + (−214. + 192i)39-s + ⋯
L(s)  = 1  + (0.998 + 0.0556i)3-s + (0.264 + 0.264i)7-s + (0.993 + 0.111i)9-s + 1.47i·11-s + (−0.836 + 0.836i)13-s + (0.468 − 0.468i)17-s − 0.338i·19-s + (0.249 + 0.278i)21-s + (1.04 + 1.04i)23-s + (0.986 + 0.166i)27-s + 0.343·29-s − 0.324·31-s + (−0.0818 + 1.46i)33-s + (0.609 + 0.609i)37-s + (−0.881 + 0.788i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.572 - 0.820i$
Analytic conductor: \(17.7005\)
Root analytic conductor: \(4.20720\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3/2),\ 0.572 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.628685129\)
\(L(\frac12)\) \(\approx\) \(2.628685129\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.18 - 0.289i)T \)
5 \( 1 \)
good7 \( 1 + (-4.89 - 4.89i)T + 343iT^{2} \)
11 \( 1 - 53.6iT - 1.33e3T^{2} \)
13 \( 1 + (39.1 - 39.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-32.8 + 32.8i)T - 4.91e3iT^{2} \)
19 \( 1 + 28iT - 6.85e3T^{2} \)
23 \( 1 + (-115. - 115. i)T + 1.21e4iT^{2} \)
29 \( 1 - 53.6T + 2.43e4T^{2} \)
31 \( 1 + 56T + 2.97e4T^{2} \)
37 \( 1 + (-137. - 137. i)T + 5.06e4iT^{2} \)
41 \( 1 + 375. iT - 6.89e4T^{2} \)
43 \( 1 + (308. - 308. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-312. + 312. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-460. - 460. i)T + 1.48e5iT^{2} \)
59 \( 1 + 375.T + 2.05e5T^{2} \)
61 \( 1 + 406T + 2.26e5T^{2} \)
67 \( 1 + (-445. - 445. i)T + 3.00e5iT^{2} \)
71 \( 1 - 751. iT - 3.57e5T^{2} \)
73 \( 1 + (215. - 215. i)T - 3.89e5iT^{2} \)
79 \( 1 + 944iT - 4.93e5T^{2} \)
83 \( 1 + (345. + 345. i)T + 5.71e5iT^{2} \)
89 \( 1 - 751.T + 7.04e5T^{2} \)
97 \( 1 + (1.19e3 + 1.19e3i)T + 9.12e5iT^{2} \)
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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

−11.61882509879031200903838968064, −10.18847588733147582119794743302, −9.513081647317220508721927832410, −8.752523940508348649135283054762, −7.42837861999784620445320356148, −7.04030549188880487398279310094, −5.14823463767289139612961206588, −4.25315164570382448301921158909, −2.77995089718173743677143334806, −1.68571223667604331300287620924, 0.929259721922078969286118544882, 2.65247603181054989394195423675, 3.61569417354218024240991301444, 4.97823448169483298766364511092, 6.30445001947033391547923084768, 7.58763476446720742895021385588, 8.251505850588629443489250547545, 9.125742320260568428836516176970, 10.24249345469381767576487308563, 10.95470348847751102676510478961

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