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Properties

Label 2-207-69.68-c5-0-36
Degree $2$
Conductor $207$
Sign $-0.974 + 0.225i$
Analytic cond. $33.1994$
Root an. cond. $5.76189$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.32i·2-s + 20.9·4-s + 58.3·5-s − 22.0i·7-s − 176. i·8-s − 194. i·10-s − 772.·11-s − 840.·13-s − 73.4·14-s + 84.6·16-s − 1.07e3·17-s − 726. i·19-s + 1.22e3·20-s + 2.56e3i·22-s + (958. − 2.34e3i)23-s + ⋯
L(s)  = 1  − 0.587i·2-s + 0.654·4-s + 1.04·5-s − 0.170i·7-s − 0.972i·8-s − 0.614i·10-s − 1.92·11-s − 1.37·13-s − 0.100·14-s + 0.0826·16-s − 0.903·17-s − 0.461i·19-s + 0.683·20-s + 1.13i·22-s + (0.377 − 0.925i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(33.1994\)
Root analytic conductor: \(5.76189\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.445259936\)
\(L(\frac12)\) \(\approx\) \(1.445259936\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + (-958. + 2.34e3i)T \)
good2 \( 1 + 3.32iT - 32T^{2} \)
5 \( 1 - 58.3T + 3.12e3T^{2} \)
7 \( 1 + 22.0iT - 1.68e4T^{2} \)
11 \( 1 + 772.T + 1.61e5T^{2} \)
13 \( 1 + 840.T + 3.71e5T^{2} \)
17 \( 1 + 1.07e3T + 1.41e6T^{2} \)
19 \( 1 + 726. iT - 2.47e6T^{2} \)
29 \( 1 + 8.09e3iT - 2.05e7T^{2} \)
31 \( 1 - 60.1T + 2.86e7T^{2} \)
37 \( 1 + 2.59e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.38e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.59e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.43e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.87e4T + 4.18e8T^{2} \)
59 \( 1 + 3.10e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.04e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.58e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.79e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.15e4T + 2.07e9T^{2} \)
79 \( 1 - 5.32e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.14e5T + 3.93e9T^{2} \)
89 \( 1 + 3.60e4T + 5.58e9T^{2} \)
97 \( 1 + 1.34e5iT - 8.58e9T^{2} \)
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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

−10.87505111822972521094676184995, −10.22511713882808014205205859190, −9.511497632888978521101151332123, −7.940595065980892663264814095129, −6.94351040830848863436661176501, −5.80047904647027395016574572538, −4.63383330638866912536238232872, −2.60209501651730104486419983646, −2.27517624641027874889542955045, −0.36103887934007597091427995212, 1.98206250382430156954169314004, 2.76770076118312728901290089000, 5.13925755024946637265119316377, 5.60884036632223334889667636921, 6.95457847318238309109206212967, 7.69320199347092457462587250603, 8.926817077373133901012954514539, 10.16346889568332243204152266258, 10.75045094901225991877294253654, 12.04801344890564202442938536864

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