Properties

Label 2-300-25.19-c1-0-0
Degree $2$
Conductor $300$
Sign $0.150 - 0.988i$
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  + (−0.587 − 0.809i)3-s + (−1.74 + 1.40i)5-s + 1.57i·7-s + (−0.309 + 0.951i)9-s + (1.19 + 3.69i)11-s + (0.326 + 0.106i)13-s + (2.15 + 0.583i)15-s + (−3.56 + 4.91i)17-s + (2.98 + 2.17i)19-s + (1.27 − 0.928i)21-s + (1.32 − 0.429i)23-s + (1.06 − 4.88i)25-s + (0.951 − 0.309i)27-s + (−2.69 + 1.95i)29-s + (4.25 + 3.08i)31-s + ⋯
L(s)  = 1  + (−0.339 − 0.467i)3-s + (−0.778 + 0.627i)5-s + 0.596i·7-s + (−0.103 + 0.317i)9-s + (0.361 + 1.11i)11-s + (0.0905 + 0.0294i)13-s + (0.557 + 0.150i)15-s + (−0.865 + 1.19i)17-s + (0.685 + 0.497i)19-s + (0.278 − 0.202i)21-s + (0.275 − 0.0895i)23-s + (0.212 − 0.977i)25-s + (0.183 − 0.0594i)27-s + (−0.500 + 0.363i)29-s + (0.763 + 0.554i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.150 - 0.988i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644898 + 0.554083i\)
\(L(\frac12)\) \(\approx\) \(0.644898 + 0.554083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (1.74 - 1.40i)T \)
good7 \( 1 - 1.57iT - 7T^{2} \)
11 \( 1 + (-1.19 - 3.69i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.326 - 0.106i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.56 - 4.91i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.98 - 2.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.32 + 0.429i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.69 - 1.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.25 - 3.08i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (8.14 + 2.64i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.394 - 1.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.42iT - 43T^{2} \)
47 \( 1 + (-0.220 - 0.303i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.64 + 9.14i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.57 - 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.38 + 10.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.14 - 8.46i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-8.19 + 5.95i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-12.5 + 4.07i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.1 + 8.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.71 + 3.74i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.24 - 6.91i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.55 - 4.89i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.13189725930426306833407632970, −11.09739388649409277836039001810, −10.29950064528781690566773054777, −9.023600880200686793955872020427, −7.998338408949018335376285600722, −7.04307770548334368365388747786, −6.25694010348868797121241429944, −4.87213500790036439625146303205, −3.57759468773413806024949408201, −1.98065366445259305259048145878, 0.67013449827203266723604842466, 3.26326331827138537847195245814, 4.35462752007238874801875942075, 5.29458655347295899244487666530, 6.64970502547156233098594366938, 7.71256991568731777290238392701, 8.806797905888860184803510639130, 9.527771610371340062671879916728, 10.89098968612364717801465709154, 11.40271096865289777741180494227

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