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Properties

Label 2-300-60.23-c2-0-15
Degree $2$
Conductor $300$
Sign $-0.820 - 0.572i$
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  + (0.837 + 1.81i)2-s + (2.69 + 1.32i)3-s + (−2.59 + 3.04i)4-s + (−0.144 + 5.99i)6-s + (3.54 + 3.54i)7-s + (−7.70 − 2.16i)8-s + (5.50 + 7.12i)9-s − 16.8·11-s + (−11.0 + 4.76i)12-s + (8.64 + 8.64i)13-s + (−3.46 + 9.40i)14-s + (−2.51 − 15.8i)16-s + (9.72 + 9.72i)17-s + (−8.31 + 15.9i)18-s + 4.78·19-s + ⋯
L(s)  = 1  + (0.418 + 0.908i)2-s + (0.897 + 0.440i)3-s + (−0.649 + 0.760i)4-s + (−0.0241 + 0.999i)6-s + (0.506 + 0.506i)7-s + (−0.962 − 0.270i)8-s + (0.611 + 0.791i)9-s − 1.53·11-s + (−0.917 + 0.396i)12-s + (0.665 + 0.665i)13-s + (−0.247 + 0.671i)14-s + (−0.157 − 0.987i)16-s + (0.572 + 0.572i)17-s + (−0.462 + 0.886i)18-s + 0.251·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.820 - 0.572i$
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: \((2,\ 300,\ (\ :1),\ -0.820 - 0.572i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.720756 + 2.29273i\)
\(L(\frac12)\) \(\approx\) \(0.720756 + 2.29273i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.837 - 1.81i)T \)
3 \( 1 + (-2.69 - 1.32i)T \)
5 \( 1 \)
good7 \( 1 + (-3.54 - 3.54i)T + 49iT^{2} \)
11 \( 1 + 16.8T + 121T^{2} \)
13 \( 1 + (-8.64 - 8.64i)T + 169iT^{2} \)
17 \( 1 + (-9.72 - 9.72i)T + 289iT^{2} \)
19 \( 1 - 4.78T + 361T^{2} \)
23 \( 1 + (13.5 + 13.5i)T + 529iT^{2} \)
29 \( 1 - 14.8T + 841T^{2} \)
31 \( 1 - 14.0iT - 961T^{2} \)
37 \( 1 + (-10.1 + 10.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (17.6 - 17.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-16.2 + 16.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (-53.9 - 53.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 36.6T + 5.04e3T^{2} \)
73 \( 1 + (-12.6 - 12.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 88.4T + 6.24e3T^{2} \)
83 \( 1 + (-63.7 - 63.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + (-85.3 + 85.3i)T - 9.40e3iT^{2} \)
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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

−12.20955016553585506883253723276, −10.83117188631510487168600268850, −9.815913670471668504319099414485, −8.631368644268378924199945606876, −8.203948123678370990834792133315, −7.23681015455058660066668580993, −5.79415326054485354461006309353, −4.86580098194663276057565600233, −3.74711490442296016463656342053, −2.43209030575040113215309628099, 0.987238481480493326925029932156, 2.49173269008352278040111209527, 3.47633076919304330475905247763, 4.77397376613328441397043999891, 5.95734914831576288359345665340, 7.62489526217481225518598645131, 8.189260192237320015246262311295, 9.461660582171958028319854333364, 10.28184997074710320204448511905, 11.12852982352429213384029502511

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