Properties

Label 2-207-207.137-c1-0-18
Degree $2$
Conductor $207$
Sign $-0.708 + 0.705i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
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  + (0.0288 + 0.0166i)2-s + (−0.373 − 1.69i)3-s + (−0.999 − 1.73i)4-s + (0.0174 − 0.0551i)6-s − 0.133i·8-s + (−2.72 + 1.26i)9-s + (−2.55 + 2.33i)12-s + (−3.20 − 5.54i)13-s + (−1.99 + 3.45i)16-s + (−0.0997 − 0.00888i)18-s + (4.15 − 2.39i)23-s + (−0.225 + 0.0498i)24-s + (2.5 − 4.33i)25-s − 0.213i·26-s + (3.15 + 4.12i)27-s + ⋯
L(s)  = 1  + (0.0204 + 0.0117i)2-s + (−0.215 − 0.976i)3-s + (−0.499 − 0.865i)4-s + (0.00711 − 0.0224i)6-s − 0.0471i·8-s + (−0.906 + 0.421i)9-s + (−0.737 + 0.674i)12-s + (−0.888 − 1.53i)13-s + (−0.499 + 0.864i)16-s + (−0.0235 − 0.00209i)18-s + (0.866 − 0.499i)23-s + (−0.0460 + 0.0101i)24-s + (0.5 − 0.866i)25-s − 0.0419i·26-s + (0.606 + 0.794i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.708 + 0.705i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ -0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322725 - 0.780894i\)
\(L(\frac12)\) \(\approx\) \(0.322725 - 0.780894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.373 + 1.69i)T \)
23 \( 1 + (-4.15 + 2.39i)T \)
good2 \( 1 + (-0.0288 - 0.0166i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.20 + 5.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
29 \( 1 + (-6.98 - 4.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.11 + 1.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-10.2 + 5.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.35 - 1.35i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (13.1 - 7.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.73iT - 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−12.35985244674343686397126500707, −10.92290871596917404895368267678, −10.26693931535903181258333772768, −8.974715439976709427203129603546, −7.968948375766825979581053959813, −6.81872142018932921747197556510, −5.73309325686285871439590257863, −4.83139003081239877312239435332, −2.66912111058446730087231552980, −0.76783635244988486525049489827, 2.93656388874325875636065785146, 4.24974489090032241071974054129, 4.99651986745256443962998769670, 6.63137077871797721655005783930, 7.86777490978264861656182656269, 9.167561338012677207336111006041, 9.484073632627789024914600465886, 10.93105554393145927645492213037, 11.76186033943076664445849077201, 12.58390884980160674755593702219

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