Properties

Label 2-300-12.11-c1-0-23
Degree $2$
Conductor $300$
Sign $0.110 + 0.993i$
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  + (1.17 − 0.780i)2-s + (−1.51 + 0.848i)3-s + (0.780 − 1.84i)4-s + (−1.11 + 2.17i)6-s − 3.02i·7-s + (−0.516 − 2.78i)8-s + (1.56 − 2.56i)9-s − 1.32·11-s + (0.382 + 3.44i)12-s + 5.12·13-s + (−2.35 − 3.56i)14-s + (−2.78 − 2.87i)16-s − 2i·17-s + (−0.158 − 4.23i)18-s − 1.32i·19-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)2-s + (−0.871 + 0.489i)3-s + (0.390 − 0.920i)4-s + (−0.456 + 0.889i)6-s − 1.14i·7-s + (−0.182 − 0.983i)8-s + (0.520 − 0.853i)9-s − 0.399·11-s + (0.110 + 0.993i)12-s + 1.42·13-s + (−0.630 − 0.951i)14-s + (−0.695 − 0.718i)16-s − 0.485i·17-s + (−0.0374 − 0.999i)18-s − 0.303i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15417 - 1.03307i\)
\(L(\frac12)\) \(\approx\) \(1.15417 - 1.03307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.780i)T \)
3 \( 1 + (1.51 - 0.848i)T \)
5 \( 1 \)
good7 \( 1 + 3.02iT - 7T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 1.32iT - 19T^{2} \)
23 \( 1 - 0.371T + 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 - 4.71iT - 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 - 7.73iT - 43T^{2} \)
47 \( 1 - 3.02T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + 4.34iT - 67T^{2} \)
71 \( 1 + 3.39T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 + 8.10iT - 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.37322350519567213477611741103, −10.74039560232277892652826018185, −10.20287699958123065425384378148, −8.999079707261716256282494779966, −7.24028958102532479004651994047, −6.35538263924068519824575907287, −5.28431288016760838679428698690, −4.29584253173765608915211451174, −3.35621918157467965496976573047, −1.07838627622742878055959292716, 2.17596323524311128721225686552, 3.85823842921416065547660384256, 5.31220684544641650870305082249, 5.88454904558691989480645404344, 6.74551305057028987246536200450, 7.983371522542226013442311084443, 8.742424477736436642054486288316, 10.41687627316335975887985570989, 11.45665198057408352252726551329, 11.98493974508743832886436001244

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