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Properties

Label 2-297-11.9-c1-0-15
Degree $2$
Conductor $297$
Sign $-0.977 - 0.209i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
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  + (0.760 − 2.33i)2-s + (−3.27 − 2.38i)4-s + (0.305 + 0.941i)5-s + (−3.49 − 2.54i)7-s + (−4.08 + 2.96i)8-s + 2.43·10-s + (−2.79 − 1.79i)11-s + (1.11 − 3.44i)13-s + (−8.60 + 6.25i)14-s + (1.33 + 4.11i)16-s + (0.816 + 2.51i)17-s + (3.09 − 2.24i)19-s + (1.23 − 3.81i)20-s + (−6.31 + 5.16i)22-s + 3.45·23-s + ⋯
L(s)  = 1  + (0.537 − 1.65i)2-s + (−1.63 − 1.19i)4-s + (0.136 + 0.420i)5-s + (−1.32 − 0.960i)7-s + (−1.44 + 1.04i)8-s + 0.770·10-s + (−0.841 − 0.540i)11-s + (0.310 − 0.954i)13-s + (−2.29 + 1.67i)14-s + (0.333 + 1.02i)16-s + (0.197 + 0.609i)17-s + (0.710 − 0.516i)19-s + (0.277 − 0.853i)20-s + (−1.34 + 1.10i)22-s + 0.719·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.977 - 0.209i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128756 + 1.21778i\)
\(L(\frac12)\) \(\approx\) \(0.128756 + 1.21778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (2.79 + 1.79i)T \)
good2 \( 1 + (-0.760 + 2.33i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.305 - 0.941i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.49 + 2.54i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.11 + 3.44i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.816 - 2.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.09 + 2.24i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.45T + 23T^{2} \)
29 \( 1 + (-8.43 - 6.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.521 - 1.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.61 + 1.90i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.63 + 6.99i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + (3.47 - 2.52i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.93 - 9.01i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.23 + 3.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.48 + 4.57i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.854T + 67T^{2} \)
71 \( 1 + (1.16 + 3.59i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.22 + 6.70i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.89 + 5.84i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.03 - 3.19i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (0.114 - 0.353i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.88360989669789838806429531717, −10.64141901561914141308348605577, −9.935500547845237537912878256927, −8.824748417203406506806681245103, −7.29770387097364163701267114085, −6.02662723674304731980861616502, −4.77166742951789992299972012913, −3.29941997388607422360805962274, −2.97993429208041549795589173535, −0.789618503585429600531718733264, 2.97923826374449544766265962161, 4.54549397969033710184319591346, 5.47354476888599995590004367095, 6.33730576274570055065583844582, 7.15792048768301588599855616355, 8.285222582057597562692508722402, 9.185840235995270712607733231044, 9.885856950116507332473529169056, 11.72201089996568427089943781967, 12.70810338537380037424880259988

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