Properties

Label 2-300-15.14-c2-0-4
Degree $2$
Conductor $300$
Sign $0.262 - 0.964i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.262 - 0.964i$
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: \((2,\ 300,\ (\ :1),\ 0.262 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.62450 + 1.24101i\)
\(L(\frac12)\) \(\approx\) \(1.62450 + 1.24101i\)
\(L(2)\) 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

−11.76034232233492194111969248428, −10.48063159404047835227780410794, −9.804632842903338699877100487122, −9.091565624689114703353065908691, −7.81485201965005719502607699608, −7.17529506865639819115084179635, −5.50584447824850053448546274877, −4.41930504001647050265224355908, −3.38617063538297609747078151090, −1.87663528629176733379438238761, 0.996311634514038622026371343584, 2.72152468322534533510698944487, 3.67595229995746786862073002174, 5.51784158888108776012118989263, 6.37601401955329073165337762089, 7.84782260458817317202938281570, 8.141673580499715912496144123306, 9.406447619869351480275821617751, 10.18432913171211522296503438995, 11.73692759185504419270634985354

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