Properties

Label 2-300-20.3-c1-0-3
Degree $2$
Conductor $300$
Sign $-0.202 - 0.979i$
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  + (1.19 + 0.760i)2-s + (−0.707 + 0.707i)3-s + (0.844 + 1.81i)4-s + (−1.38 + 0.305i)6-s + (0.611 + 0.611i)7-s + (−0.371 + 2.80i)8-s − 1.00i·9-s + 5.12i·11-s + (−1.87 − 0.685i)12-s + (−1.76 − 1.76i)13-s + (0.264 + 1.19i)14-s + (−2.57 + 3.06i)16-s + (3.76 − 3.76i)17-s + (0.760 − 1.19i)18-s + 1.22·19-s + ⋯
L(s)  = 1  + (0.843 + 0.537i)2-s + (−0.408 + 0.408i)3-s + (0.422 + 0.906i)4-s + (−0.563 + 0.124i)6-s + (0.231 + 0.231i)7-s + (−0.131 + 0.991i)8-s − 0.333i·9-s + 1.54i·11-s + (−0.542 − 0.197i)12-s + (−0.488 − 0.488i)13-s + (0.0706 + 0.319i)14-s + (−0.643 + 0.765i)16-s + (0.912 − 0.912i)17-s + (0.179 − 0.281i)18-s + 0.280·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13750 + 1.39682i\)
\(L(\frac12)\) \(\approx\) \(1.13750 + 1.39682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.760i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-0.611 - 0.611i)T + 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + (-3.76 + 3.76i)T - 17iT^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (1.07 - 1.07i)T - 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 + 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (-6.20 + 6.20i)T - 43iT^{2} \)
47 \( 1 + (-2.29 - 2.29i)T + 47iT^{2} \)
53 \( 1 + (-2.62 - 2.62i)T + 53iT^{2} \)
59 \( 1 - 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (6.20 + 6.20i)T + 67iT^{2} \)
71 \( 1 + 8.10iT - 71T^{2} \)
73 \( 1 + (-2.25 - 2.25i)T + 73iT^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (7.95 - 7.95i)T - 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (0.793 - 0.793i)T - 97iT^{2} \)
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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

−12.15752499777811838882742710865, −11.38248678805688350230969922354, −10.13889347768842235719976583752, −9.266897164347256520560218133726, −7.75496785758455594209066838188, −7.17501766360477705980692444291, −5.79041397785364009713854360786, −5.02255926388856189343434774233, −4.03987637093437876168409431052, −2.50397301425774830549780813618, 1.22331731144603691286806814467, 2.93765893526049545430800879575, 4.20932516906761187634339787704, 5.50433221346294074822234844370, 6.21133704013261044874207117177, 7.39501081159020148585789376439, 8.621303086232019050096094542816, 9.982269154285210913090019628963, 10.85138375295746761599568879437, 11.54748365596167495822374814750

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