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Properties

Label 2-300-75.8-c1-0-9
Degree $2$
Conductor $300$
Sign $-0.645 + 0.764i$
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  + (−0.0384 − 1.73i)3-s + (−2.22 − 0.176i)5-s + (3.29 − 3.29i)7-s + (−2.99 + 0.132i)9-s + (−2.51 + 0.817i)11-s + (−2.61 + 1.33i)13-s + (−0.219 + 3.86i)15-s + (−1.07 − 6.77i)17-s + (0.761 − 1.04i)19-s + (−5.83 − 5.58i)21-s + (0.460 + 0.234i)23-s + (4.93 + 0.786i)25-s + (0.345 + 5.18i)27-s + (3.38 − 2.46i)29-s + (−6.95 − 5.05i)31-s + ⋯
L(s)  = 1  + (−0.0221 − 0.999i)3-s + (−0.996 − 0.0788i)5-s + (1.24 − 1.24i)7-s + (−0.999 + 0.0443i)9-s + (−0.758 + 0.246i)11-s + (−0.725 + 0.369i)13-s + (−0.0567 + 0.998i)15-s + (−0.260 − 1.64i)17-s + (0.174 − 0.240i)19-s + (−1.27 − 1.21i)21-s + (0.0959 + 0.0488i)23-s + (0.987 + 0.157i)25-s + (0.0664 + 0.997i)27-s + (0.629 − 0.457i)29-s + (−1.24 − 0.907i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.645 + 0.764i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.645 + 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405040 - 0.872178i\)
\(L(\frac12)\) \(\approx\) \(0.405040 - 0.872178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0384 + 1.73i)T \)
5 \( 1 + (2.22 + 0.176i)T \)
good7 \( 1 + (-3.29 + 3.29i)T - 7iT^{2} \)
11 \( 1 + (2.51 - 0.817i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.61 - 1.33i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.07 + 6.77i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-0.761 + 1.04i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.460 - 0.234i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (-3.38 + 2.46i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.95 + 5.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.868 - 1.70i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-6.79 - 2.20i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-6.48 - 6.48i)T + 43iT^{2} \)
47 \( 1 + (-9.43 - 1.49i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.368 + 2.32i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-0.216 + 0.664i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.56 + 4.80i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.0 + 1.74i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-3.02 - 4.16i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.39 - 2.74i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-1.41 - 1.94i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.525 + 0.0832i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-1.78 - 5.48i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.58 + 10.0i)T + (-92.2 - 29.9i)T^{2} \)
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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

−11.34350660150566014821651453430, −10.95868669212755703774849207005, −9.420571821122323936728132241306, −8.062320333266126681658364918659, −7.53448512799744002097765182621, −7.00478724414987546697823205388, −5.17140872460950028229249825521, −4.29637428865373382536913907773, −2.53661717787635095692372329926, −0.71888028650778590459696379365, 2.51323850794350614679763154832, 3.90704177557408387370582835043, 5.02024391264836837319339459132, 5.74164410693706159000757157069, 7.59911082770871085892357661849, 8.422955730534324164497766128874, 9.007211277521307156465327381070, 10.62677014024148824695397613784, 10.88161028291015432350917774068, 12.04703347259618735011333720825

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