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Properties

Label 2-297-9.4-c1-0-9
Degree $2$
Conductor $297$
Sign $-0.173 - 0.984i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (−1 + 1.73i)5-s + (−2 − 3.46i)7-s + 3.99·10-s + (0.5 + 0.866i)11-s + (−2 + 3.46i)13-s + (−3.99 + 6.92i)14-s + (1.99 + 3.46i)16-s − 4·17-s − 6·19-s + (−2 − 3.46i)20-s + (0.999 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (0.500 + 0.866i)25-s + 7.99·26-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.447 + 0.774i)5-s + (−0.755 − 1.30i)7-s + 1.26·10-s + (0.150 + 0.261i)11-s + (−0.554 + 0.960i)13-s + (−1.06 + 1.85i)14-s + (0.499 + 0.866i)16-s − 0.970·17-s − 1.37·19-s + (−0.447 − 0.774i)20-s + (0.213 − 0.369i)22-s + (−0.104 + 0.180i)23-s + (0.100 + 0.173i)25-s + 1.56·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.173 - 0.984i$
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: \(1\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.95499757473882603226545618130, −10.29395986250559392884437524957, −9.551826926631181505926788507992, −8.509153422029926724631821535134, −7.10375162472758260745881929549, −6.56871821402738322683438601724, −4.30360202497684370705698568529, −3.43677556435135175114633298612, −2.05933520908841347903091490138, 0, 2.78998953712282322401398440323, 4.67100900698936684166211889573, 5.86337034063811022888697183384, 6.52581389050606359161365143536, 7.84516899732185259191929311161, 8.634684866455110705691323555663, 9.100048405657324587845127126321, 10.18183933596119276234822335504, 11.63743641857147060960167169102

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