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Properties

Label 2-297-9.4-c1-0-8
Degree $2$
Conductor $297$
Sign $-0.574 + 0.818i$
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  + (−0.447 − 0.774i)2-s + (0.599 − 1.03i)4-s + (1.87 − 3.24i)5-s + (−0.725 − 1.25i)7-s − 2.86·8-s − 3.35·10-s + (0.5 + 0.866i)11-s + (−2.87 + 4.98i)13-s + (−0.648 + 1.12i)14-s + (0.0800 + 0.138i)16-s + 4.79·17-s + 0.702·19-s + (−2.24 − 3.89i)20-s + (0.447 − 0.774i)22-s + (−0.825 + 1.42i)23-s + ⋯
L(s)  = 1  + (−0.316 − 0.547i)2-s + (0.299 − 0.519i)4-s + (0.838 − 1.45i)5-s + (−0.274 − 0.474i)7-s − 1.01·8-s − 1.06·10-s + (0.150 + 0.261i)11-s + (−0.798 + 1.38i)13-s + (−0.173 + 0.300i)14-s + (0.0200 + 0.0346i)16-s + 1.16·17-s + 0.161·19-s + (−0.502 − 0.871i)20-s + (0.0953 − 0.165i)22-s + (−0.172 + 0.298i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570305 - 1.09640i\)
\(L(\frac12)\) \(\approx\) \(0.570305 - 1.09640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.447 + 0.774i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.87 + 3.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.725 + 1.25i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.87 - 4.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 - 0.702T + 19T^{2} \)
23 \( 1 + (0.825 - 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.15 + 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.65 + 2.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.05 - 3.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.898 - 1.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.15T + 53T^{2} \)
59 \( 1 + (2.32 - 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.27 + 2.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.47 - 7.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.14T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + (-0.543 - 0.941i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.90 + 3.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.01T + 89T^{2} \)
97 \( 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.58357200105452096063785751810, −10.19544062952930763581190543143, −9.559685488212701184602713693685, −9.124416906972712347404951305417, −7.65564798060893007523382565338, −6.31970981770695910610426691197, −5.40222224762515483883103816957, −4.25319714860678780290281860183, −2.25559590501399052094013641507, −1.07451238539708379326247017686, 2.63559281160481689643556930549, 3.24778954966398264087508307882, 5.57988247599024176754321459805, 6.25287255892737190834240967030, 7.27594911438221528518038910586, 7.987023074283492710115804285065, 9.325058591834241195530835341475, 10.13779653122239788965071397644, 10.99991286736205155472883661142, 12.13423741500482662341447166568

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