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Properties

Label 2-297-9.7-c1-0-9
Degree $2$
Conductor $297$
Sign $-0.995 + 0.0952i$
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  + (1.23 − 2.13i)2-s + (−2.04 − 3.54i)4-s + (−1.21 − 2.10i)5-s + (−1.16 + 2.02i)7-s − 5.17·8-s − 6.01·10-s + (0.5 − 0.866i)11-s + (−2.35 − 4.07i)13-s + (2.88 + 5.00i)14-s + (−2.29 + 3.97i)16-s + 3.20·17-s + 7.77·19-s + (−4.99 + 8.64i)20-s + (−1.23 − 2.13i)22-s + (1.37 + 2.38i)23-s + ⋯
L(s)  = 1  + (0.873 − 1.51i)2-s + (−1.02 − 1.77i)4-s + (−0.544 − 0.943i)5-s + (−0.441 + 0.765i)7-s − 1.83·8-s − 1.90·10-s + (0.150 − 0.261i)11-s + (−0.653 − 1.13i)13-s + (0.771 + 1.33i)14-s + (−0.574 + 0.994i)16-s + 0.778·17-s + 1.78·19-s + (−1.11 + 1.93i)20-s + (−0.263 − 0.455i)22-s + (0.287 + 0.498i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.995 + 0.0952i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.995 + 0.0952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0767269 - 1.60728i\)
\(L(\frac12)\) \(\approx\) \(0.0767269 - 1.60728i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.21 + 2.10i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (2.35 + 4.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.20T + 17T^{2} \)
19 \( 1 - 7.77T + 19T^{2} \)
23 \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.18 - 2.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.685 - 1.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + (-1.77 - 3.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.103 + 0.180i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.11T + 53T^{2} \)
59 \( 1 + (0.120 + 0.208i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.830 - 1.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.07T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.25 + 9.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (4.46 - 7.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.80834150020566232356596325406, −10.53361613519318076199126312286, −9.677427667971238422677525148427, −8.835267447289611209144487558525, −7.54919990708069963868226981601, −5.51763868808239874760977225673, −5.16415529824442774697746845658, −3.68817235418356270794278228526, −2.81791468141550656424330555331, −0.996987043021249174463446750216, 3.24044559876913570430302556662, 4.13779592021100690038884342785, 5.29926962075759967517274762263, 6.57999560490173060278455742902, 7.21899980838273558642323443219, 7.69493527596541145864624108589, 9.192165043078166794492796342066, 10.26715858555976127739568101807, 11.58130503378748994499006761150, 12.36954767341878001280737071467

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