Properties

Label 2-300-4.3-c2-0-7
Degree $2$
Conductor $300$
Sign $-0.652 - 0.757i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.87 + 0.696i)2-s + 1.73i·3-s + (3.02 − 2.61i)4-s + (−1.20 − 3.24i)6-s − 5.46i·7-s + (−3.86 + 7.00i)8-s − 2.99·9-s + 11.0i·11-s + (4.52 + 5.24i)12-s − 10.1·13-s + (3.80 + 10.2i)14-s + (2.35 − 15.8i)16-s + 24.4·17-s + (5.62 − 2.08i)18-s + 23.7i·19-s + ⋯
L(s)  = 1  + (−0.937 + 0.348i)2-s + 0.577i·3-s + (0.757 − 0.652i)4-s + (−0.201 − 0.541i)6-s − 0.781i·7-s + (−0.482 + 0.875i)8-s − 0.333·9-s + 1.00i·11-s + (0.376 + 0.437i)12-s − 0.778·13-s + (0.272 + 0.732i)14-s + (0.147 − 0.989i)16-s + 1.43·17-s + (0.312 − 0.116i)18-s + 1.25i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.652 - 0.757i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.652 - 0.757i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.311851 + 0.680598i\)
\(L(\frac12)\) \(\approx\) \(0.311851 + 0.680598i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 - 0.696i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 + 5.46iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 + 10.1T + 169T^{2} \)
17 \( 1 - 24.4T + 289T^{2} \)
19 \( 1 - 23.7iT - 361T^{2} \)
23 \( 1 - 37.2iT - 529T^{2} \)
29 \( 1 + 25.7T + 841T^{2} \)
31 \( 1 + 4.83iT - 961T^{2} \)
37 \( 1 + 35.6T + 1.36e3T^{2} \)
41 \( 1 + 9.30T + 1.68e3T^{2} \)
43 \( 1 - 70.0iT - 1.84e3T^{2} \)
47 \( 1 - 38.0iT - 2.20e3T^{2} \)
53 \( 1 + 55.7T + 2.80e3T^{2} \)
59 \( 1 - 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 - 76.7iT - 5.04e3T^{2} \)
73 \( 1 - 93.5T + 5.32e3T^{2} \)
79 \( 1 + 49.3iT - 6.24e3T^{2} \)
83 \( 1 + 72.3iT - 6.88e3T^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 - 72.9T + 9.40e3T^{2} \)
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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

−11.64804796159972554974986643881, −10.53297691517109449464682282396, −9.876891244180425250369548957118, −9.348758789335094981315371759407, −7.73449956921370032826419279608, −7.50024178529091049494812742895, −6.00909300018571437243018757774, −4.93646776777897967380199884982, −3.43467498262371742651199555536, −1.57939099203242462227164767656, 0.49484728645730302719364501048, 2.21806415804789340045358297050, 3.28019375946473253548931554382, 5.33679400231403273060460407094, 6.49905229362027337743507333896, 7.47879797880192651766998477262, 8.474277739671944668002032592793, 9.108249722022896309444834061108, 10.25608517084672215896607425250, 11.17087121635146803873204440199

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