Properties

Label 2-300-15.14-c2-0-8
Degree $2$
Conductor $300$
Sign $-0.956 + 0.291i$
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  + (−2.95 − 0.5i)3-s + 8i·7-s + (8.5 + 2.95i)9-s − 17.7i·11-s − 2i·13-s − 17.7·17-s − 11·19-s + (4 − 23.6i)21-s − 35.4·23-s + (−23.6 − 13i)27-s + 35.4i·29-s − 46·31-s + (−8.87 + 52.5i)33-s − 16i·37-s + (−1 + 5.91i)39-s + ⋯
L(s)  = 1  + (−0.986 − 0.166i)3-s + 1.14i·7-s + (0.944 + 0.328i)9-s − 1.61i·11-s − 0.153i·13-s − 1.04·17-s − 0.578·19-s + (0.190 − 1.12i)21-s − 1.54·23-s + (−0.876 − 0.481i)27-s + 1.22i·29-s − 1.48·31-s + (−0.268 + 1.59i)33-s − 0.432i·37-s + (−0.0256 + 0.151i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.956 + 0.291i$
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: \((2,\ 300,\ (\ :1),\ -0.956 + 0.291i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0238742 - 0.160023i\)
\(L(\frac12)\) \(\approx\) \(0.0238742 - 0.160023i\)
\(L(2)\) 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 \( 1 + (2.95 + 0.5i)T \)
5 \( 1 \)
good7 \( 1 - 8iT - 49T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 + 2iT - 169T^{2} \)
17 \( 1 + 17.7T + 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + 35.4T + 529T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 + 46T + 961T^{2} \)
37 \( 1 + 16iT - 1.36e3T^{2} \)
41 \( 1 + 53.2iT - 1.68e3T^{2} \)
43 \( 1 + 62iT - 1.84e3T^{2} \)
47 \( 1 + 35.4T + 2.20e3T^{2} \)
53 \( 1 - 35.4T + 2.80e3T^{2} \)
59 \( 1 + 70.9iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 - 113iT - 4.48e3T^{2} \)
71 \( 1 - 106. iT - 5.04e3T^{2} \)
73 \( 1 + 101iT - 5.32e3T^{2} \)
79 \( 1 + 68T + 6.24e3T^{2} \)
83 \( 1 - 17.7T + 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 + 22iT - 9.40e3T^{2} \)
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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

−11.13898917366130498728645162184, −10.48481631442826818526972236734, −9.070924380074935629513078403014, −8.372934043276650756535241827613, −6.95942774827552603533961551296, −5.90526927732681388990142879591, −5.39008185217622609256893890710, −3.82282415323021516604790355313, −2.09128915811338465236388886178, −0.085071757099238205817910951762, 1.80903108124915669021394032373, 4.14065181655108974557940821923, 4.60609002384745704697562082634, 6.14906916111516207060160817382, 6.98575358375508349243388827645, 7.84356027685349237923237774835, 9.522155116802746718360919871937, 10.13395717292301154632788795569, 10.98274475009191848179014790726, 11.85112931274378496631924742633

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