Properties

Label 2-300-20.19-c2-0-24
Degree $2$
Conductor $300$
Sign $-0.999 + 0.0401i$
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  + (−1.08 − 1.67i)2-s − 1.73·3-s + (−1.64 + 3.64i)4-s + (1.87 + 2.90i)6-s + 0.596·7-s + (7.91 − 1.19i)8-s + 2.99·9-s + 9.27i·11-s + (2.84 − 6.31i)12-s − 23.5i·13-s + (−0.647 − 1.00i)14-s + (−10.5 − 11.9i)16-s − 3.97i·17-s + (−3.25 − 5.03i)18-s + 7.04i·19-s + ⋯
L(s)  = 1  + (−0.542 − 0.839i)2-s − 0.577·3-s + (−0.410 + 0.911i)4-s + (0.313 + 0.484i)6-s + 0.0852·7-s + (0.988 − 0.149i)8-s + 0.333·9-s + 0.843i·11-s + (0.237 − 0.526i)12-s − 1.80i·13-s + (−0.0462 − 0.0715i)14-s + (−0.662 − 0.749i)16-s − 0.233i·17-s + (−0.180 − 0.279i)18-s + 0.370i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.999 + 0.0401i$
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: \((2,\ 300,\ (\ :1),\ -0.999 + 0.0401i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00820418 - 0.408814i\)
\(L(\frac12)\) \(\approx\) \(0.00820418 - 0.408814i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 1.67i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 - 0.596T + 49T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 + 23.5iT - 169T^{2} \)
17 \( 1 + 3.97iT - 289T^{2} \)
19 \( 1 - 7.04iT - 361T^{2} \)
23 \( 1 + 32.0T + 529T^{2} \)
29 \( 1 + 35.6T + 841T^{2} \)
31 \( 1 + 59.2iT - 961T^{2} \)
37 \( 1 - 5.38iT - 1.36e3T^{2} \)
41 \( 1 - 40.0T + 1.68e3T^{2} \)
43 \( 1 + 36.1T + 1.84e3T^{2} \)
47 \( 1 + 74.0T + 2.20e3T^{2} \)
53 \( 1 + 2.55iT - 2.80e3T^{2} \)
59 \( 1 + 36.4iT - 3.48e3T^{2} \)
61 \( 1 + 8.73T + 3.72e3T^{2} \)
67 \( 1 - 69.7T + 4.48e3T^{2} \)
71 \( 1 + 59.2iT - 5.04e3T^{2} \)
73 \( 1 + 83.0iT - 5.32e3T^{2} \)
79 \( 1 + 65.8iT - 6.24e3T^{2} \)
83 \( 1 + 129.T + 6.88e3T^{2} \)
89 \( 1 - 130.T + 7.92e3T^{2} \)
97 \( 1 + 93.1iT - 9.40e3T^{2} \)
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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

−11.06216065509057297865828860111, −10.07899986613063208570106987668, −9.617486739119830394519076025482, −8.081461610290299811878943795817, −7.56063884858972284678670145335, −5.99592163964137142706714917112, −4.77912856142762448735957758025, −3.52251686373217960187460707348, −1.98704329632274732566181784947, −0.25770002253266983709640297545, 1.62688537757731947559648723965, 4.05124651232778023108469142368, 5.20765275237015429492367852930, 6.27634177813643042935962645526, 6.96385380375212400001050727214, 8.178316059334748283469934256438, 9.072808299959063640418077085723, 9.954001622468304805194404610950, 11.04388093208420473558831289415, 11.67512075021026758828939775357

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