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Properties

Label 2-300-12.11-c1-0-25
Degree $2$
Conductor $300$
Sign $-0.566 + 0.824i$
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  + (−0.599 + 1.28i)2-s + (−0.468 − 1.66i)3-s + (−1.28 − 1.53i)4-s + (2.41 + 0.400i)6-s − 0.936i·7-s + (2.73 − 0.719i)8-s + (−2.56 + 1.56i)9-s − 4.27·11-s + (−1.96 + 2.85i)12-s − 3.12·13-s + (1.19 + 0.561i)14-s + (−0.719 + 3.93i)16-s − 2i·17-s + (−0.463 − 4.21i)18-s − 4.27i·19-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (−0.270 − 0.962i)3-s + (−0.640 − 0.768i)4-s + (0.986 + 0.163i)6-s − 0.353i·7-s + (0.967 − 0.254i)8-s + (−0.853 + 0.520i)9-s − 1.28·11-s + (−0.566 + 0.824i)12-s − 0.866·13-s + (0.320 + 0.150i)14-s + (−0.179 + 0.983i)16-s − 0.485i·17-s + (−0.109 − 0.994i)18-s − 0.979i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.566 + 0.824i$
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: \((2,\ 300,\ (\ :1/2),\ -0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173307 - 0.329374i\)
\(L(\frac12)\) \(\approx\) \(0.173307 - 0.329374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 - 1.28i)T \)
3 \( 1 + (0.468 + 1.66i)T \)
5 \( 1 \)
good7 \( 1 + 0.936iT - 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4.27iT - 19T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 - 0.936T + 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 + 5.20iT - 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 - 9.06iT - 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
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

−11.37078416809400436548166559455, −10.35903322841464958423255742732, −9.442136116203789078910832880450, −8.049309674512852047592161169015, −7.66119027476579965686524176123, −6.66539556658376431598065448425, −5.64651560968772193466329559823, −4.66104511109969010822380245253, −2.36764872704517405538761170340, −0.30608622156097079726339614810, 2.36519375137095092997258297346, 3.62676430646168778544878298472, 4.79841076686813089934800056249, 5.78525142880518885569640046816, 7.65014392703888512268570491526, 8.511929255288157834020332559266, 9.548403868939200975405928098983, 10.32515300610893261235417179891, 10.80905031174912895067488697493, 12.09742652096447331577419305170

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