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Properties

Label 2-300-60.23-c2-0-48
Degree $2$
Conductor $300$
Sign $0.819 + 0.573i$
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 + (2.78 + 1.12i)3-s + (2.81 − 2.84i)4-s + (5.99 − 0.0674i)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 − 4.74i)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 + (16.7 + 6.56i)18-s + 1.48·19-s + ⋯
L(s)  = 1  + (0.922 − 0.385i)2-s + (0.927 + 0.375i)3-s + (0.702 − 0.711i)4-s + (0.999 − 0.0112i)6-s + (−0.678 − 0.678i)7-s + (0.374 − 0.927i)8-s + (0.718 + 0.695i)9-s + 1.08·11-s + (0.918 − 0.395i)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 + (0.931 + 0.364i)18-s + 0.0783·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.819 + 0.573i$
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: \((2,\ 300,\ (\ :1),\ 0.819 + 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.54757 - 1.11749i\)
\(L(\frac12)\) \(\approx\) \(3.54757 - 1.11749i\)
\(L(2)\) 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 \( 1 + (-2.78 - 1.12i)T \)
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

−11.45426334491181886581345991641, −10.53773830416155110617970493226, −9.631665900665190597617896623658, −8.866641635559862962764448650372, −7.24130237185560793880041477699, −6.61187233436374023207342394979, −5.02847214224354357739035747488, −3.89640355185350388235907564740, −3.22744555301919056515079127603, −1.60808519751934363000999728295, 2.02101611907275886038315892815, 3.27893883198492412026399609949, 4.17518349130964124486539148244, 5.83951724674713513009668348478, 6.61282439367073886287588199401, 7.59714322618728505367473431785, 8.718275650506203030082558647177, 9.377791913609099604070678459412, 10.90367829059379101840423050685, 12.01806120434370339049101433482

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