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Properties

Label 2-297-9.4-c1-0-2
Degree $2$
Conductor $297$
Sign $0.766 - 0.642i$
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 − 1.73i)4-s + (−1.5 + 2.59i)5-s + (2 + 3.46i)7-s + (0.5 + 0.866i)11-s + (−1 + 1.73i)13-s + (−1.99 − 3.46i)16-s + 6·17-s + 2·19-s + (3 + 5.19i)20-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + 7.99·28-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s − 12·35-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.670 + 1.16i)5-s + (0.755 + 1.30i)7-s + (0.150 + 0.261i)11-s + (−0.277 + 0.480i)13-s + (−0.499 − 0.866i)16-s + 1.45·17-s + 0.458·19-s + (0.670 + 1.16i)20-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + 1.51·28-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s − 2.02·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.766 - 0.642i$
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: \((2,\ 297,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29952 + 0.472990i\)
\(L(\frac12)\) \(\approx\) \(1.29952 + 0.472990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.70976840224178657645255304602, −11.06604717898899859312658473439, −10.13285004614896473390972268123, −9.143899373925332674364886818872, −7.84407403359628160909987312779, −6.99411788708267933737730978084, −5.91933809565732762678394874742, −4.96750753260414997550522412980, −3.18895724760629212409882567965, −1.95645976579469289711952343714, 1.18288470684720682365807688112, 3.40337086461650862084060922825, 4.29692155081749853412246097558, 5.45222262379005889363150632727, 7.26346311317985676759872702551, 7.74964898218221678158799336440, 8.479493175686953196774443449814, 9.766138619873537607499634856535, 11.02416648215562685265395079788, 11.64283768159508618847728196554

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