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Properties

Label 2-300-60.23-c2-0-47
Degree $2$
Conductor $300$
Sign $0.939 + 0.341i$
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  + (1.99 + 0.141i)2-s + (2.06 − 2.17i)3-s + (3.95 + 0.565i)4-s + (4.43 − 4.04i)6-s + (5.18 + 5.18i)7-s + (7.81 + 1.68i)8-s + (−0.459 − 8.98i)9-s − 7.14·11-s + (9.41 − 7.44i)12-s + (7.93 + 7.93i)13-s + (9.61 + 11.0i)14-s + (15.3 + 4.47i)16-s + (−16.5 − 16.5i)17-s + (0.357 − 17.9i)18-s − 12.1·19-s + ⋯
L(s)  = 1  + (0.997 + 0.0708i)2-s + (0.688 − 0.724i)3-s + (0.989 + 0.141i)4-s + (0.738 − 0.674i)6-s + (0.741 + 0.741i)7-s + (0.977 + 0.211i)8-s + (−0.0510 − 0.998i)9-s − 0.649·11-s + (0.784 − 0.620i)12-s + (0.610 + 0.610i)13-s + (0.686 + 0.791i)14-s + (0.960 + 0.279i)16-s + (−0.975 − 0.975i)17-s + (0.0198 − 0.999i)18-s − 0.639·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.939 + 0.341i$
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: \((2,\ 300,\ (\ :1),\ 0.939 + 0.341i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.81662 - 0.671221i\)
\(L(\frac12)\) \(\approx\) \(3.81662 - 0.671221i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.141i)T \)
3 \( 1 + (-2.06 + 2.17i)T \)
5 \( 1 \)
good7 \( 1 + (-5.18 - 5.18i)T + 49iT^{2} \)
11 \( 1 + 7.14T + 121T^{2} \)
13 \( 1 + (-7.93 - 7.93i)T + 169iT^{2} \)
17 \( 1 + (16.5 + 16.5i)T + 289iT^{2} \)
19 \( 1 + 12.1T + 361T^{2} \)
23 \( 1 + (-11.0 - 11.0i)T + 529iT^{2} \)
29 \( 1 + 26.1T + 841T^{2} \)
31 \( 1 + 8.74iT - 961T^{2} \)
37 \( 1 + (-26.7 + 26.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (24.6 - 24.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (58.6 - 58.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (20.4 - 20.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-35.8 - 35.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 46.2T + 5.04e3T^{2} \)
73 \( 1 + (10.6 + 10.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 68.3T + 6.24e3T^{2} \)
83 \( 1 + (-76.6 - 76.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 41.0T + 7.92e3T^{2} \)
97 \( 1 + (81.7 - 81.7i)T - 9.40e3iT^{2} \)
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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

−11.54849940509804855088143456862, −11.08528874193579402445550965938, −9.369725734805666765669902574603, −8.404900258890325298573685192130, −7.48816955527137603625327078887, −6.52880562279669519060079299303, −5.44483619915971711718603656774, −4.22945424817747451664734283462, −2.78362754049091931678782287118, −1.82896069702694944037099599906, 1.94136427635981391889724440431, 3.34706099814052034624415443016, 4.34138287301247561505044864287, 5.16357240425482632831871671863, 6.53053805623002049065839078791, 7.83437575942815839034508553378, 8.482030332607810840509518429691, 10.06611280634307754718347137497, 10.79968669926078473902470442758, 11.28766493046486823194715384572

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