Properties

Label 2-300-100.67-c1-0-18
Degree $2$
Conductor $300$
Sign $0.916 + 0.400i$
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  + (−1.37 − 0.315i)2-s + (0.987 + 0.156i)3-s + (1.80 + 0.869i)4-s + (2.22 + 0.203i)5-s + (−1.31 − 0.527i)6-s + (2.91 − 2.91i)7-s + (−2.20 − 1.76i)8-s + (0.951 + 0.309i)9-s + (−3.00 − 0.982i)10-s + (−2.79 + 0.909i)11-s + (1.64 + 1.14i)12-s + (−0.922 − 1.81i)13-s + (−4.93 + 3.09i)14-s + (2.16 + 0.549i)15-s + (2.48 + 3.13i)16-s + (0.653 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.974 − 0.223i)2-s + (0.570 + 0.0903i)3-s + (0.900 + 0.434i)4-s + (0.995 + 0.0909i)5-s + (−0.535 − 0.215i)6-s + (1.10 − 1.10i)7-s + (−0.780 − 0.624i)8-s + (0.317 + 0.103i)9-s + (−0.950 − 0.310i)10-s + (−0.843 + 0.274i)11-s + (0.474 + 0.329i)12-s + (−0.255 − 0.502i)13-s + (−1.31 + 0.827i)14-s + (0.559 + 0.141i)15-s + (0.621 + 0.783i)16-s + (0.158 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23919 - 0.258948i\)
\(L(\frac12)\) \(\approx\) \(1.23919 - 0.258948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.315i)T \)
3 \( 1 + (-0.987 - 0.156i)T \)
5 \( 1 + (-2.22 - 0.203i)T \)
good7 \( 1 + (-2.91 + 2.91i)T - 7iT^{2} \)
11 \( 1 + (2.79 - 0.909i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.922 + 1.81i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.653 - 4.12i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.04 + 1.48i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.577 + 1.13i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (4.10 + 5.65i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.56 - 6.28i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.09 + 4.63i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.48 - 4.56i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-4.56 - 4.56i)T + 43iT^{2} \)
47 \( 1 + (0.753 - 4.75i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.396 + 2.50i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.41 - 4.34i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.01 - 9.27i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (14.9 - 2.36i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-3.33 - 4.58i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (7.97 + 4.06i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (7.70 - 5.60i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.36 + 14.9i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.68 + 1.19i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (17.8 + 2.82i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.18210605768073799380507211401, −10.52677817506802325132598848344, −9.959258442970377910318474067880, −8.853489326115811203883180039710, −7.896433213968866922875249502736, −7.28149196353799492594823500260, −5.87124870674345445273034545202, −4.35733571873914228593724408157, −2.69655203418185935459627741100, −1.50408334670466798406493683199, 1.81606470498647717376982698914, 2.63788817332231409692386699278, 5.10308904754891714813269816571, 5.87097573608176853060357344194, 7.24405554601224017313398256974, 8.165206147775521640883068361151, 9.016013592476608098302717110369, 9.582811380376105501641248233038, 10.72865789151859422095288162462, 11.57101963970000836515702076885

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