Properties

Label 2-207-207.11-c1-0-18
Degree $2$
Conductor $207$
Sign $0.783 + 0.621i$
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  + (2.16 − 0.102i)2-s + (−0.197 − 1.72i)3-s + (2.66 − 0.254i)4-s + (0.848 + 0.809i)5-s + (−0.602 − 3.69i)6-s + (0.0842 + 0.437i)7-s + (1.45 − 0.209i)8-s + (−2.92 + 0.678i)9-s + (1.91 + 1.66i)10-s + (−1.13 + 0.583i)11-s + (−0.963 − 4.53i)12-s + (−1.03 − 0.198i)13-s + (0.227 + 0.935i)14-s + (1.22 − 1.62i)15-s + (−2.14 + 0.412i)16-s + (−0.577 − 1.26i)17-s + ⋯
L(s)  = 1  + (1.52 − 0.0727i)2-s + (−0.113 − 0.993i)3-s + (1.33 − 0.127i)4-s + (0.379 + 0.361i)5-s + (−0.246 − 1.50i)6-s + (0.0318 + 0.165i)7-s + (0.513 − 0.0738i)8-s + (−0.974 + 0.226i)9-s + (0.606 + 0.525i)10-s + (−0.341 + 0.176i)11-s + (−0.278 − 1.31i)12-s + (−0.285 − 0.0550i)13-s + (0.0606 + 0.250i)14-s + (0.316 − 0.418i)15-s + (−0.535 + 0.103i)16-s + (−0.139 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28481 - 0.796387i\)
\(L(\frac12)\) \(\approx\) \(2.28481 - 0.796387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.197 + 1.72i)T \)
23 \( 1 + (-0.998 - 4.69i)T \)
good2 \( 1 + (-2.16 + 0.102i)T + (1.99 - 0.190i)T^{2} \)
5 \( 1 + (-0.848 - 0.809i)T + (0.237 + 4.99i)T^{2} \)
7 \( 1 + (-0.0842 - 0.437i)T + (-6.49 + 2.60i)T^{2} \)
11 \( 1 + (1.13 - 0.583i)T + (6.38 - 8.96i)T^{2} \)
13 \( 1 + (1.03 + 0.198i)T + (12.0 + 4.83i)T^{2} \)
17 \( 1 + (0.577 + 1.26i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-4.62 - 2.11i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.365 + 3.83i)T + (-28.4 - 5.48i)T^{2} \)
31 \( 1 + (4.02 - 3.16i)T + (7.30 - 30.1i)T^{2} \)
37 \( 1 + (-1.33 + 4.53i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (5.08 - 5.33i)T + (-1.95 - 40.9i)T^{2} \)
43 \( 1 + (-0.532 + 0.677i)T + (-10.1 - 41.7i)T^{2} \)
47 \( 1 + (-0.449 - 0.259i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.31 + 1.51i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (1.22 - 6.38i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (-2.97 + 7.43i)T + (-44.1 - 42.0i)T^{2} \)
67 \( 1 + (-5.40 + 10.4i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (6.34 + 9.87i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (-4.30 + 9.41i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-13.4 + 4.64i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (5.98 - 5.71i)T + (3.94 - 82.9i)T^{2} \)
89 \( 1 + (1.75 - 12.2i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (5.86 + 1.42i)T + (86.2 + 44.4i)T^{2} \)
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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

−12.37220231742185335839758860493, −11.83211250188728135118977424714, −10.83005558247402365130828135614, −9.393845085884055422491330855736, −7.84362711773281988655091665172, −6.84931227097772798998448706562, −5.86442898436036913228807184990, −5.05554047161507687644482115391, −3.33032547187784215766214417981, −2.17072487229543263351542348028, 2.81705711401151934833932569723, 4.00807996277430334447215974448, 5.06642301107106837272127346065, 5.67241492347455452712308889211, 7.00343705600889280278591795033, 8.683428485831900054665380116621, 9.681685024019818415254346192135, 10.82646270620720895832478152650, 11.66532117030099699314961749268, 12.67738967875419386417967630966

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