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Properties

Label 2-207-23.22-c6-0-57
Degree $2$
Conductor $207$
Sign $-0.988 - 0.150i$
Analytic cond. $47.6211$
Root an. cond. $6.90081$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.6·2-s + 49.0·4-s − 66.6i·5-s − 306. i·7-s − 159.·8-s − 708. i·10-s − 998. i·11-s − 1.93e3·13-s − 3.26e3i·14-s − 4.83e3·16-s + 9.46e3i·17-s + 9.74e3i·19-s − 3.26e3i·20-s − 1.06e4i·22-s + (−1.20e4 − 1.83e3i)23-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.765·4-s − 0.532i·5-s − 0.894i·7-s − 0.311·8-s − 0.708i·10-s − 0.750i·11-s − 0.878·13-s − 1.18i·14-s − 1.17·16-s + 1.92i·17-s + 1.42i·19-s − 0.408i·20-s − 0.997i·22-s + (−0.988 − 0.150i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(47.6211\)
Root analytic conductor: \(6.90081\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :3),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.6158807347\)
\(L(\frac12)\) \(\approx\) \(0.6158807347\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + (1.20e4 + 1.83e3i)T \)
good2 \( 1 - 10.6T + 64T^{2} \)
5 \( 1 + 66.6iT - 1.56e4T^{2} \)
7 \( 1 + 306. iT - 1.17e5T^{2} \)
11 \( 1 + 998. iT - 1.77e6T^{2} \)
13 \( 1 + 1.93e3T + 4.82e6T^{2} \)
17 \( 1 - 9.46e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.74e3iT - 4.70e7T^{2} \)
29 \( 1 + 4.41e4T + 5.94e8T^{2} \)
31 \( 1 - 1.29e4T + 8.87e8T^{2} \)
37 \( 1 + 9.38e4iT - 2.56e9T^{2} \)
41 \( 1 + 3.57e4T + 4.75e9T^{2} \)
43 \( 1 + 674. iT - 6.32e9T^{2} \)
47 \( 1 + 1.02e5T + 1.07e10T^{2} \)
53 \( 1 + 1.85e5iT - 2.21e10T^{2} \)
59 \( 1 + 7.62e4T + 4.21e10T^{2} \)
61 \( 1 - 4.05e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.14e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.12e5T + 1.28e11T^{2} \)
73 \( 1 + 1.20e5T + 1.51e11T^{2} \)
79 \( 1 + 3.85e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.30e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 - 2.70e4iT - 8.32e11T^{2} \)
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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

−10.99335912566390422510783010836, −10.03802285903317574009185673856, −8.696200116339738576151081083863, −7.63089236254998770374869380016, −6.23326996852457525087215121916, −5.45408284736400596682792737915, −4.16516456733737986741594849286, −3.58723657166302702769523888099, −1.81870763986128988856372586624, −0.093706332985268217177576707580, 2.35437014179445229706352223695, 3.04430983608046329618963431444, 4.65348057163564662477317496938, 5.24649000324945631268700236840, 6.55114467783345130964598298507, 7.38256748812814738212375013156, 9.050388349765837496514590443725, 9.816860125753993741265045574349, 11.42867537975123389264709955457, 11.87332251732036313013442016185

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