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Properties

Label 2-300-75.2-c1-0-6
Degree $2$
Conductor $300$
Sign $0.754 + 0.656i$
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  + (0.194 − 1.72i)3-s + (2.22 − 0.237i)5-s + (2.44 + 2.44i)7-s + (−2.92 − 0.668i)9-s + (0.626 + 0.862i)11-s + (4.64 − 0.735i)13-s + (0.0218 − 3.87i)15-s + (−5.63 − 2.87i)17-s + (−2.86 + 0.932i)19-s + (4.68 − 3.73i)21-s + (−4.44 − 0.703i)23-s + (4.88 − 1.05i)25-s + (−1.71 + 4.90i)27-s + (2.43 − 7.49i)29-s + (−0.586 − 1.80i)31-s + ⋯
L(s)  = 1  + (0.112 − 0.993i)3-s + (0.994 − 0.106i)5-s + (0.923 + 0.923i)7-s + (−0.974 − 0.222i)9-s + (0.188 + 0.259i)11-s + (1.28 − 0.203i)13-s + (0.00565 − 0.999i)15-s + (−1.36 − 0.696i)17-s + (−0.658 + 0.213i)19-s + (1.02 − 0.814i)21-s + (−0.926 − 0.146i)23-s + (0.977 − 0.211i)25-s + (−0.330 + 0.943i)27-s + (0.452 − 1.39i)29-s + (−0.105 − 0.324i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.754 + 0.656i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53489 - 0.574099i\)
\(L(\frac12)\) \(\approx\) \(1.53489 - 0.574099i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.194 + 1.72i)T \)
5 \( 1 + (-2.22 + 0.237i)T \)
good7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 + (-0.626 - 0.862i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-4.64 + 0.735i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (5.63 + 2.87i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (2.86 - 0.932i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.44 + 0.703i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-2.43 + 7.49i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.586 + 1.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.993 - 6.27i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-1.22 + 1.69i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (8.27 - 8.27i)T - 43iT^{2} \)
47 \( 1 + (-4.80 - 9.42i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (9.25 - 4.71i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.79 + 2.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.42 + 1.75i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.94 + 5.77i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (0.855 + 0.277i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.277 - 1.75i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (8.30 + 2.69i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.42 - 4.76i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-1.94 + 1.41i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.23 - 2.15i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.65174644735307239771378903352, −10.97550531241371109193358398524, −9.541931758587347200414588057339, −8.627995802519008540218908655074, −8.013135816837352222765878268246, −6.43209241948928996070557185738, −5.99714904910777197938231444269, −4.65892298235088646319092016563, −2.56907152714950799334720177282, −1.62775733425093254338463868269, 1.86028702724538919400302858460, 3.68117781520137751145983202703, 4.61440046833733984135005041405, 5.79585271072633548066579677561, 6.81280641515604381802571066080, 8.475477496139008921168295897404, 8.881589432366270716397116161291, 10.24626877447625461343400381745, 10.75134128021459255941769515376, 11.38621699770171608800366984058

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