Properties

Label 2-207-69.68-c5-0-21
Degree $2$
Conductor $207$
Sign $-0.721 + 0.692i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.84i·2-s − 46.3·4-s − 32.1·5-s + 32.0i·7-s + 126. i·8-s + 284. i·10-s + 601.·11-s + 865.·13-s + 284.·14-s − 360.·16-s + 831.·17-s + 1.07e3i·19-s + 1.48e3·20-s − 5.32e3i·22-s + (2.49e3 + 478. i)23-s + ⋯
L(s)  = 1  − 1.56i·2-s − 1.44·4-s − 0.574·5-s + 0.247i·7-s + 0.700i·8-s + 0.898i·10-s + 1.49·11-s + 1.42·13-s + 0.387·14-s − 0.352·16-s + 0.697·17-s + 0.681i·19-s + 0.831·20-s − 2.34i·22-s + (0.982 + 0.188i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.721 + 0.692i$
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: \((2,\ 207,\ (\ :5/2),\ -0.721 + 0.692i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.908110494\)
\(L(\frac12)\) \(\approx\) \(1.908110494\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.49e3 - 478. i)T \)
good2 \( 1 + 8.84iT - 32T^{2} \)
5 \( 1 + 32.1T + 3.12e3T^{2} \)
7 \( 1 - 32.0iT - 1.68e4T^{2} \)
11 \( 1 - 601.T + 1.61e5T^{2} \)
13 \( 1 - 865.T + 3.71e5T^{2} \)
17 \( 1 - 831.T + 1.41e6T^{2} \)
19 \( 1 - 1.07e3iT - 2.47e6T^{2} \)
29 \( 1 - 1.74e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.61e3T + 2.86e7T^{2} \)
37 \( 1 + 1.45e4iT - 6.93e7T^{2} \)
41 \( 1 + 4.60e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.61e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.93e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.14e4T + 4.18e8T^{2} \)
59 \( 1 + 2.14e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.06e3iT - 8.44e8T^{2} \)
67 \( 1 - 6.90e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.05e3iT - 1.80e9T^{2} \)
73 \( 1 - 7.80e3T + 2.07e9T^{2} \)
79 \( 1 + 1.87e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.01e4T + 3.93e9T^{2} \)
89 \( 1 - 2.98e4T + 5.58e9T^{2} \)
97 \( 1 + 1.26e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.28818022841696207963693145541, −10.47313092380185468210398561169, −9.265595039325328500776959986277, −8.659430367708777459353983839771, −7.12428687869942589222986010327, −5.69311279451906174266852677323, −3.91056145585212017088329881085, −3.57302474913260642543415958529, −1.82502097697206610346010232199, −0.78511785179983801937705922113, 1.04682383138423641593573594249, 3.57859347930695654570401099398, 4.62469085249369528154231475203, 6.01059902879000428018766068051, 6.73044855301566940891313932299, 7.72636172996635166483516346374, 8.636876043627911385791298650906, 9.432527139879026332874296820938, 11.04211316723851544242665592086, 11.82547796404642029624034763643

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