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Properties

Label 2-207-23.22-c6-0-17
Degree $2$
Conductor $207$
Sign $0.961 - 0.274i$
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  + 3.69·2-s − 50.3·4-s − 218. i·5-s + 261. i·7-s − 422.·8-s − 808. i·10-s + 1.77e3i·11-s − 2.88e3·13-s + 965. i·14-s + 1.66e3·16-s − 559. i·17-s + 1.59e3i·19-s + 1.10e4i·20-s + 6.54e3i·22-s + (1.16e4 − 3.34e3i)23-s + ⋯
L(s)  = 1  + 0.461·2-s − 0.786·4-s − 1.75i·5-s + 0.762i·7-s − 0.824·8-s − 0.808i·10-s + 1.33i·11-s − 1.31·13-s + 0.351i·14-s + 0.405·16-s − 0.113i·17-s + 0.232i·19-s + 1.37i·20-s + 0.614i·22-s + (0.961 − 0.274i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.961 - 0.274i$
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: \((2,\ 207,\ (\ :3),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.513778322\)
\(L(\frac12)\) \(\approx\) \(1.513778322\)
\(L(4)\) 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 \( 1 + (-1.16e4 + 3.34e3i)T \)
good2 \( 1 - 3.69T + 64T^{2} \)
5 \( 1 + 218. iT - 1.56e4T^{2} \)
7 \( 1 - 261. iT - 1.17e5T^{2} \)
11 \( 1 - 1.77e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.88e3T + 4.82e6T^{2} \)
17 \( 1 + 559. iT - 2.41e7T^{2} \)
19 \( 1 - 1.59e3iT - 4.70e7T^{2} \)
29 \( 1 - 1.99e4T + 5.94e8T^{2} \)
31 \( 1 - 1.44e4T + 8.87e8T^{2} \)
37 \( 1 + 6.33e4iT - 2.56e9T^{2} \)
41 \( 1 + 4.34e3T + 4.75e9T^{2} \)
43 \( 1 + 2.64e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.55e5T + 1.07e10T^{2} \)
53 \( 1 - 1.19e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.35e5T + 4.21e10T^{2} \)
61 \( 1 - 2.10e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.94e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.08e5T + 1.28e11T^{2} \)
73 \( 1 + 3.93e4T + 1.51e11T^{2} \)
79 \( 1 - 1.50e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.26e5iT - 3.26e11T^{2} \)
89 \( 1 + 5.76e5iT - 4.96e11T^{2} \)
97 \( 1 + 9.73e5iT - 8.32e11T^{2} \)
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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

−12.08482048116052946839216186631, −10.03293935413096482165480338591, −9.220627184752826794989732924227, −8.673511044138154683533019790743, −7.43168523973383451647333643428, −5.65748419401611924574614752299, −4.88664426487595884523549768391, −4.30841013151366507572613145104, −2.38495591593035890757967384322, −0.816545467107764408531662511037, 0.51129040515885019438657053936, 2.79946608167555063085590514460, 3.50326870417072887927112881828, 4.82279820384856620136944149425, 6.14583474106961031313029360100, 7.05601257458306041553382468186, 8.127155450726824310999766299301, 9.504700474909660401739190398838, 10.40784653374580413557629187564, 11.15987860784168570027633512519

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