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Properties

Label 2-300-60.23-c2-0-2
Degree $2$
Conductor $300$
Sign $-0.157 - 0.987i$
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  + (−1.84 + 0.770i)2-s + (−1.12 − 2.78i)3-s + (2.81 − 2.84i)4-s + (4.21 + 4.26i)6-s + (−4.75 − 4.75i)7-s + (−2.99 + 7.41i)8-s + (−6.46 + 6.25i)9-s − 11.9·11-s + (−11.0 − 4.61i)12-s + (4.22 + 4.22i)13-s + (12.4 + 5.10i)14-s + (−0.188 − 15.9i)16-s + (9.35 + 9.35i)17-s + (7.11 − 16.5i)18-s + 1.48·19-s + ⋯
L(s)  = 1  + (−0.922 + 0.385i)2-s + (−0.375 − 0.927i)3-s + (0.702 − 0.711i)4-s + (0.703 + 0.710i)6-s + (−0.678 − 0.678i)7-s + (−0.374 + 0.927i)8-s + (−0.718 + 0.695i)9-s − 1.08·11-s + (−0.922 − 0.384i)12-s + (0.324 + 0.324i)13-s + (0.887 + 0.364i)14-s + (−0.0117 − 0.999i)16-s + (0.550 + 0.550i)17-s + (0.395 − 0.918i)18-s + 0.0783·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.185619 + 0.217639i\)
\(L(\frac12)\) \(\approx\) \(0.185619 + 0.217639i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.84 - 0.770i)T \)
3 \( 1 + (1.12 + 2.78i)T \)
5 \( 1 \)
good7 \( 1 + (4.75 + 4.75i)T + 49iT^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 + (-4.22 - 4.22i)T + 169iT^{2} \)
17 \( 1 + (-9.35 - 9.35i)T + 289iT^{2} \)
19 \( 1 - 1.48T + 361T^{2} \)
23 \( 1 + (11.6 + 11.6i)T + 529iT^{2} \)
29 \( 1 - 39.3T + 841T^{2} \)
31 \( 1 - 43.6iT - 961T^{2} \)
37 \( 1 + (49.1 - 49.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 27.0iT - 1.68e3T^{2} \)
43 \( 1 + (8.84 - 8.84i)T - 1.84e3iT^{2} \)
47 \( 1 + (15.0 - 15.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-14.6 + 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 61.7iT - 3.48e3T^{2} \)
61 \( 1 + 84.6T + 3.72e3T^{2} \)
67 \( 1 + (-65.7 - 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 14.2T + 5.04e3T^{2} \)
73 \( 1 + (16.0 + 16.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 9.32T + 6.24e3T^{2} \)
83 \( 1 + (12.7 + 12.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.4T + 7.92e3T^{2} \)
97 \( 1 + (6.90 - 6.90i)T - 9.40e3iT^{2} \)
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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

−11.71696832616117322687194518639, −10.49134208103555107584492990849, −10.19339886552583936070864723061, −8.624636405494183795725321405610, −7.945775846806737538207495496610, −6.92041719504597937998667007504, −6.29129027089331084935438454368, −5.09869487842903732411234966633, −2.91129851859364736183562918807, −1.30377610579393086147262164335, 0.20587686582550664789800644373, 2.60509145120115666141525287183, 3.63079520189280562640230485914, 5.30307297118469702402596119646, 6.26624902064442992648773970697, 7.66342122955264343073580538696, 8.686283558533516634519218715849, 9.565354749340405936967823645623, 10.22084256602068028479632723470, 11.01419023643263834916930384021

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