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Properties

Label 2-207-69.68-c5-0-8
Degree $2$
Conductor $207$
Sign $-0.0328 + 0.999i$
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  − 10.4i·2-s − 77.1·4-s − 72.3·5-s − 9.31i·7-s + 471. i·8-s + 755. i·10-s − 308.·11-s + 245.·13-s − 97.3·14-s + 2.46e3·16-s + 762.·17-s + 2.84e3i·19-s + 5.58e3·20-s + 3.22e3i·22-s + (−2.02e3 − 1.53e3i)23-s + ⋯
L(s)  = 1  − 1.84i·2-s − 2.41·4-s − 1.29·5-s − 0.0718i·7-s + 2.60i·8-s + 2.39i·10-s − 0.769·11-s + 0.403·13-s − 0.132·14-s + 2.40·16-s + 0.639·17-s + 1.81i·19-s + 3.12·20-s + 1.42i·22-s + (−0.797 − 0.603i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0328 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0328 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.0328 + 0.999i$
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: \((2,\ 207,\ (\ :5/2),\ -0.0328 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7670651304\)
\(L(\frac12)\) \(\approx\) \(0.7670651304\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.02e3 + 1.53e3i)T \)
good2 \( 1 + 10.4iT - 32T^{2} \)
5 \( 1 + 72.3T + 3.12e3T^{2} \)
7 \( 1 + 9.31iT - 1.68e4T^{2} \)
11 \( 1 + 308.T + 1.61e5T^{2} \)
13 \( 1 - 245.T + 3.71e5T^{2} \)
17 \( 1 - 762.T + 1.41e6T^{2} \)
19 \( 1 - 2.84e3iT - 2.47e6T^{2} \)
29 \( 1 + 6.02e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.57e3T + 2.86e7T^{2} \)
37 \( 1 + 9.44e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.27e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.41e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.76e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.43e4T + 4.18e8T^{2} \)
59 \( 1 + 5.42e3iT - 7.14e8T^{2} \)
61 \( 1 + 3.79e3iT - 8.44e8T^{2} \)
67 \( 1 + 2.35e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.94e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.09e4T + 2.07e9T^{2} \)
79 \( 1 - 8.82e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.34e4T + 3.93e9T^{2} \)
89 \( 1 + 4.89e4T + 5.58e9T^{2} \)
97 \( 1 - 7.15e4iT - 8.58e9T^{2} \)
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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

−11.30902526095384550276078690285, −10.52676098339438186149626416724, −9.717937637795798635064039006963, −8.363033760267273945728710700609, −7.76679052150272830981460803626, −5.58987033531222377588582373242, −4.11202331978902356896711348392, −3.59974086883409208702229759139, −2.17560888981415619265177886053, −0.66223190669040774961090642775, 0.41997792089696758698389065138, 3.47372830919768198248534671875, 4.66161864977699390999685482835, 5.57858783215018916013073255122, 6.92772525289328991617806137180, 7.58450803863282830859005834491, 8.393414745888632247803396358190, 9.253408466846867543876739570229, 10.70360130684121300316822585802, 11.93780391555702527453650361377

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