Properties

Label 2-300-300.119-c1-0-0
Degree $2$
Conductor $300$
Sign $0.291 - 0.956i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.129 − 1.40i)2-s + (−1.59 − 0.677i)3-s + (−1.96 + 0.363i)4-s + (−1.18 − 1.89i)5-s + (−0.748 + 2.33i)6-s − 2.57·7-s + (0.766 + 2.72i)8-s + (2.08 + 2.16i)9-s + (−2.51 + 1.91i)10-s + (0.341 + 1.04i)11-s + (3.38 + 0.753i)12-s + (2.85 + 0.927i)13-s + (0.332 + 3.62i)14-s + (0.603 + 3.82i)15-s + (3.73 − 1.43i)16-s + (−2.78 − 2.02i)17-s + ⋯
L(s)  = 1  + (−0.0913 − 0.995i)2-s + (−0.920 − 0.391i)3-s + (−0.983 + 0.181i)4-s + (−0.529 − 0.848i)5-s + (−0.305 + 0.952i)6-s − 0.971·7-s + (0.271 + 0.962i)8-s + (0.693 + 0.720i)9-s + (−0.796 + 0.605i)10-s + (0.102 + 0.316i)11-s + (0.976 + 0.217i)12-s + (0.791 + 0.257i)13-s + (0.0887 + 0.967i)14-s + (0.155 + 0.987i)15-s + (0.933 − 0.357i)16-s + (−0.675 − 0.490i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.291 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0327887 + 0.0242822i\)
\(L(\frac12)\) \(\approx\) \(0.0327887 + 0.0242822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.129 + 1.40i)T \)
3 \( 1 + (1.59 + 0.677i)T \)
5 \( 1 + (1.18 + 1.89i)T \)
good7 \( 1 + 2.57T + 7T^{2} \)
11 \( 1 + (-0.341 - 1.04i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.85 - 0.927i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.78 + 2.02i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.92 - 2.65i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.10 - 1.33i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.76 - 2.42i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.24 - 5.83i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.09 + 1.97i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.47 + 0.478i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + (6.37 + 8.78i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.5 - 7.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.55 + 10.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.600 - 1.84i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.29 - 0.937i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (9.09 - 6.60i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.87 - 3.20i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.915 + 1.26i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.91 + 5.38i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (9.75 - 3.17i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.79 - 7.97i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.00206059443343115103171857843, −11.15516076486800599249256771600, −10.25841628028667028686894010585, −9.259559008367381289938345932303, −8.347727941389767222564566106525, −7.07249414481805271217083359877, −5.80761041139480430933915990907, −4.66416250020997657969019354745, −3.63519912428678183711183529193, −1.64297205455285196319084705632, 0.03541412396306411754807764680, 3.51641583825517020952379566095, 4.43683887507566798675096536183, 6.10540866434099449298687659175, 6.31404982050636735951885296716, 7.41989029610742757234217614902, 8.640343566838786297960004559245, 9.724237310175510948476746905829, 10.56064844474421003943291838701, 11.35490668300377119288862663064

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