Properties

Label 2-300-20.19-c2-0-25
Degree $2$
Conductor $300$
Sign $0.245 + 0.969i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.696 + 1.87i)2-s − 1.73·3-s + (−3.02 + 2.61i)4-s + (−1.20 − 3.24i)6-s − 5.46·7-s + (−7.00 − 3.86i)8-s + 2.99·9-s + 11.0i·11-s + (5.24 − 4.52i)12-s − 10.1i·13-s + (−3.80 − 10.2i)14-s + (2.35 − 15.8i)16-s − 24.4i·17-s + (2.08 + 5.62i)18-s − 23.7i·19-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s − 0.577·3-s + (−0.757 + 0.652i)4-s + (−0.201 − 0.541i)6-s − 0.781·7-s + (−0.875 − 0.482i)8-s + 0.333·9-s + 1.00i·11-s + (0.437 − 0.376i)12-s − 0.778i·13-s + (−0.272 − 0.732i)14-s + (0.147 − 0.989i)16-s − 1.43i·17-s + (0.116 + 0.312i)18-s − 1.25i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.260122 - 0.202499i\)
\(L(\frac12)\) \(\approx\) \(0.260122 - 0.202499i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.696 - 1.87i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 + 5.46T + 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 + 10.1iT - 169T^{2} \)
17 \( 1 + 24.4iT - 289T^{2} \)
19 \( 1 + 23.7iT - 361T^{2} \)
23 \( 1 + 37.2T + 529T^{2} \)
29 \( 1 - 25.7T + 841T^{2} \)
31 \( 1 + 4.83iT - 961T^{2} \)
37 \( 1 - 35.6iT - 1.36e3T^{2} \)
41 \( 1 + 9.30T + 1.68e3T^{2} \)
43 \( 1 + 70.0T + 1.84e3T^{2} \)
47 \( 1 - 38.0T + 2.20e3T^{2} \)
53 \( 1 + 55.7iT - 2.80e3T^{2} \)
59 \( 1 + 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 + 104.T + 4.48e3T^{2} \)
71 \( 1 - 76.7iT - 5.04e3T^{2} \)
73 \( 1 - 93.5iT - 5.32e3T^{2} \)
79 \( 1 - 49.3iT - 6.24e3T^{2} \)
83 \( 1 - 72.3T + 6.88e3T^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + 72.9iT - 9.40e3T^{2} \)
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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

−11.64500414659030846019391786533, −10.08239290309522400273814367491, −9.536496288352725704542464426352, −8.226408350076543032076635920155, −7.12760724169411075939370776300, −6.49915021223464300090618530699, −5.31269944459667891178265516542, −4.46181605059323958566011779550, −2.95855493521653830754323066245, −0.15229773930998085731103407902, 1.67340402599160487299100163761, 3.40073686725094139516549949627, 4.29276919790792870034772401809, 5.87542257654585922439013317934, 6.26436790866478454026310135046, 8.106089807667550653639338988098, 9.143733809072750690517542606539, 10.24612794665116996387853089078, 10.68121428999581049308598811834, 11.97332282025293580573512307888

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