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Properties

Label 2-297-11.4-c1-0-3
Degree $2$
Conductor $297$
Sign $0.865 - 0.500i$
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  + (−0.984 − 0.715i)2-s + (−0.160 − 0.494i)4-s + (−0.240 + 0.174i)5-s + (1.32 + 4.08i)7-s + (−0.947 + 2.91i)8-s + 0.361·10-s + (−2.91 + 1.58i)11-s + (3.06 + 2.22i)13-s + (1.61 − 4.97i)14-s + (2.17 − 1.58i)16-s + (1.09 − 0.796i)17-s + (1.01 − 3.12i)19-s + (0.124 + 0.0907i)20-s + (4.00 + 0.516i)22-s + 4.75·23-s + ⋯
L(s)  = 1  + (−0.695 − 0.505i)2-s + (−0.0803 − 0.247i)4-s + (−0.107 + 0.0780i)5-s + (0.501 + 1.54i)7-s + (−0.334 + 1.03i)8-s + 0.114·10-s + (−0.877 + 0.479i)11-s + (0.849 + 0.616i)13-s + (0.431 − 1.32i)14-s + (0.544 − 0.395i)16-s + (0.265 − 0.193i)17-s + (0.233 − 0.717i)19-s + (0.0279 + 0.0202i)20-s + (0.853 + 0.110i)22-s + 0.991·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771489 + 0.206741i\)
\(L(\frac12)\) \(\approx\) \(0.771489 + 0.206741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 - 1.58i)T \)
good2 \( 1 + (0.984 + 0.715i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.240 - 0.174i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.32 - 4.08i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.06 - 2.22i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.09 + 0.796i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.01 + 3.12i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.75T + 23T^{2} \)
29 \( 1 + (-3.09 - 9.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.15 + 3.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.40 - 4.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.87 + 5.78i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.92T + 43T^{2} \)
47 \( 1 + (-1.47 + 4.55i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.16 - 0.845i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.95 - 6.01i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.41 + 6.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.68T + 67T^{2} \)
71 \( 1 + (5.17 - 3.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.511 - 1.57i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.52 + 4.01i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.08 - 2.97i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + (3.15 + 2.29i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.48682940789183208415484294556, −11.04421101684038642250251783772, −9.890853223101985435586295801236, −8.922909228783547162190756238761, −8.527592797265572361778288519719, −7.12332697045744564079837166136, −5.61154290143614638431197659922, −5.00718406927905269395364563544, −2.90606852468253521906509412838, −1.70612263911683837954246862731, 0.801072296714126626768235543596, 3.34631089723658753843937905446, 4.37983592476870187644699415692, 5.93085959464424030049380085543, 7.15297536364385771094054690243, 7.980911085781261067837332210916, 8.408095727116535602869975035515, 9.884631443706908644929403654539, 10.52110286247959738377822281883, 11.46887939393595980341423029245

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