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Properties

Label 2-300-15.14-c2-0-3
Degree $2$
Conductor $300$
Sign $0.929 - 0.368i$
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  + (−2.23 − 2i)3-s + 2i·7-s + (1.00 + 8.94i)9-s + 13.4i·11-s − 8i·13-s − 13.4·17-s + 34·19-s + (4 − 4.47i)21-s + 40.2·23-s + (15.6 − 22.0i)27-s + 40.2i·29-s + 14·31-s + (26.8 − 30.0i)33-s + 56i·37-s + (−16 + 17.8i)39-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)3-s + 0.285i·7-s + (0.111 + 0.993i)9-s + 1.21i·11-s − 0.615i·13-s − 0.789·17-s + 1.78·19-s + (0.190 − 0.212i)21-s + 1.74·23-s + (0.579 − 0.814i)27-s + 1.38i·29-s + 0.451·31-s + (0.813 − 0.909i)33-s + 1.51i·37-s + (−0.410 + 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18112 + 0.225574i\)
\(L(\frac12)\) \(\approx\) \(1.18112 + 0.225574i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.23 + 2i)T \)
5 \( 1 \)
good7 \( 1 - 2iT - 49T^{2} \)
11 \( 1 - 13.4iT - 121T^{2} \)
13 \( 1 + 8iT - 169T^{2} \)
17 \( 1 + 13.4T + 289T^{2} \)
19 \( 1 - 34T + 361T^{2} \)
23 \( 1 - 40.2T + 529T^{2} \)
29 \( 1 - 40.2iT - 841T^{2} \)
31 \( 1 - 14T + 961T^{2} \)
37 \( 1 - 56iT - 1.36e3T^{2} \)
41 \( 1 + 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 8iT - 1.84e3T^{2} \)
47 \( 1 - 40.2T + 2.20e3T^{2} \)
53 \( 1 + 40.2T + 2.80e3T^{2} \)
59 \( 1 + 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 46T + 3.72e3T^{2} \)
67 \( 1 - 32iT - 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 106iT - 5.32e3T^{2} \)
79 \( 1 - 22T + 6.24e3T^{2} \)
83 \( 1 + 120.T + 6.88e3T^{2} \)
89 \( 1 + 107. iT - 7.92e3T^{2} \)
97 \( 1 - 122iT - 9.40e3T^{2} \)
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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

−11.67491287396899532945683869913, −10.78814719538746616487173932599, −9.816994786239068141701716547466, −8.687171284818910508052261474275, −7.38465657961656225191887835176, −6.88568984473372541655652726549, −5.49875699226628949783339057997, −4.76656131635516598791009424454, −2.83405182838099226883775213756, −1.25078637838298401631685047854, 0.77475133942346317145728847786, 3.13385976447391963677200149841, 4.33337270366909505230202698452, 5.42653055467533229033956953344, 6.38054458799849057792642677908, 7.47909025193132483199793278401, 8.927268389431399089464542199731, 9.545643525936812910738848792386, 10.78807501206847077592719275585, 11.28345877427665201339789754580

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