Properties

Label 2-300-3.2-c2-0-4
Degree $2$
Conductor $300$
Sign $-0.166 - 0.986i$
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  + (0.5 + 2.95i)3-s + 8·7-s + (−8.5 + 2.95i)9-s + 17.7i·11-s + 2·13-s − 17.7i·17-s + 11·19-s + (4 + 23.6i)21-s + 35.4i·23-s + (−13 − 23.6i)27-s + 35.4i·29-s − 46·31-s + (−52.5 + 8.87i)33-s − 16·37-s + (1 + 5.91i)39-s + ⋯
L(s)  = 1  + (0.166 + 0.986i)3-s + 1.14·7-s + (−0.944 + 0.328i)9-s + 1.61i·11-s + 0.153·13-s − 1.04i·17-s + 0.578·19-s + (0.190 + 1.12i)21-s + 1.54i·23-s + (−0.481 − 0.876i)27-s + 1.22i·29-s − 1.48·31-s + (−1.59 + 0.268i)33-s − 0.432·37-s + (0.0256 + 0.151i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.166 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13557 + 1.34363i\)
\(L(\frac12)\) \(\approx\) \(1.13557 + 1.34363i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 2.95i)T \)
5 \( 1 \)
good7 \( 1 - 8T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 - 2T + 169T^{2} \)
17 \( 1 + 17.7iT - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 - 35.4iT - 529T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 + 46T + 961T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 - 53.2iT - 1.68e3T^{2} \)
43 \( 1 - 62T + 1.84e3T^{2} \)
47 \( 1 + 35.4iT - 2.20e3T^{2} \)
53 \( 1 + 35.4iT - 2.80e3T^{2} \)
59 \( 1 + 70.9iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 - 113T + 4.48e3T^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 - 101T + 5.32e3T^{2} \)
79 \( 1 - 68T + 6.24e3T^{2} \)
83 \( 1 + 17.7iT - 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 + 22T + 9.40e3T^{2} \)
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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

−11.51525246239819527911685180196, −10.93280812058074498721714801564, −9.731395012728915745321242009842, −9.235329608531963292093678103702, −7.951284140513758110233013754312, −7.14904264957317762419506239656, −5.30245628762735146094373289716, −4.82042838775270658577833779909, −3.52749662578710638260824167725, −1.90322271469735628113089084601, 0.891175805139584751911206463716, 2.32611010148222984827042867444, 3.82881755400475318390393523677, 5.48030764346161712456414394084, 6.26689384145394101632241593526, 7.57745218031872433605670467942, 8.324582313167455274400977642759, 8.959312525831639762538225788284, 10.72617755657221988164781823058, 11.22953439718119637035639662125

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