Properties

Label 2-300-3.2-c2-0-10
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

\(L(\frac{3}{2})\) \(\approx\) \(0.328579 - 0.388780i\)
\(L(\frac12)\) \(\approx\) \(0.328579 - 0.388780i\)
\(L(2)\) 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

−11.32314994118341667389154363753, −10.19739360360986183982715067682, −9.458390124849823287851288606533, −8.737652391024185121131660011102, −7.34841724765203637668411687462, −6.01549815096420120498686756195, −5.34552455857369841628635571314, −3.73806419816889662133776980963, −3.03150199046996107002501097680, −0.24284099450680643829129442202, 1.74875850009925863695828967600, 3.13388297968388083664314454966, 4.75754471503577270226583644173, 6.14507178889110193025756837554, 6.86860326362973904427192882929, 7.72153956458941178226485922323, 8.938785858761975445950802467361, 9.916747465624891889209550840271, 10.84661852940972404532680115808, 12.15765111793053226969972882269

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