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Properties

Label 2-297-11.9-c1-0-14
Degree $2$
Conductor $297$
Sign $-0.999 + 0.00104i$
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  + (0.725 − 2.23i)2-s + (−2.84 − 2.06i)4-s + (−1.06 − 3.26i)5-s + (2.30 + 1.67i)7-s + (−2.88 + 2.09i)8-s − 8.06·10-s + (−3.10 − 1.16i)11-s + (0.236 − 0.729i)13-s + (5.41 − 3.93i)14-s + (0.416 + 1.28i)16-s + (1.17 + 3.60i)17-s + (−2.30 + 1.67i)19-s + (−3.73 + 11.4i)20-s + (−4.85 + 6.09i)22-s + 8.97·23-s + ⋯
L(s)  = 1  + (0.513 − 1.57i)2-s + (−1.42 − 1.03i)4-s + (−0.474 − 1.46i)5-s + (0.871 + 0.633i)7-s + (−1.02 + 0.741i)8-s − 2.55·10-s + (−0.936 − 0.351i)11-s + (0.0657 − 0.202i)13-s + (1.44 − 1.05i)14-s + (0.104 + 0.320i)16-s + (0.283 + 0.873i)17-s + (−0.528 + 0.383i)19-s + (−0.834 + 2.56i)20-s + (−1.03 + 1.29i)22-s + 1.87·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.999 + 0.00104i$
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: \((2,\ 297,\ (\ :1/2),\ -0.999 + 0.00104i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000766753 - 1.47000i\)
\(L(\frac12)\) \(\approx\) \(0.000766753 - 1.47000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.10 + 1.16i)T \)
good2 \( 1 + (-0.725 + 2.23i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.06 + 3.26i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.30 - 1.67i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.236 + 0.729i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.17 - 3.60i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.30 - 1.67i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.97T + 23T^{2} \)
29 \( 1 + (1.90 + 1.38i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.86 + 8.81i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.67 - 2.66i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.71 - 2.69i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (-3.08 + 2.24i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.93 + 5.94i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.15 + 0.840i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.169 - 0.522i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.14T + 67T^{2} \)
71 \( 1 + (-3.14 - 9.69i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-13.2 - 9.64i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.890 + 2.73i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.500 - 1.54i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.72T + 89T^{2} \)
97 \( 1 + (3.19 - 9.84i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.41502396631066549995551598487, −10.73823939534458093440492966083, −9.558264692034794862174871783499, −8.591347214018420356666161575536, −7.932903222084400801433385612538, −5.59174382002163252335681781534, −4.89773382440022991832354790715, −3.95510153304999305277214411107, −2.42985808516046706152203118329, −1.02682830160423836138894731866, 2.96418551214824277790585328675, 4.40083815136688764218245571134, 5.26368549886822378565698452572, 6.66155055888804315619631116071, 7.30142859410819641262585687572, 7.77971856816369778726741819634, 9.025780422012823691347824954828, 10.69493959485339898085918939386, 10.99490256598186034180499029874, 12.46645575313306885439130704165

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