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Properties

Label 2-207-69.68-c5-0-38
Degree $2$
Conductor $207$
Sign $0.581 - 0.813i$
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  − 7.94i·2-s − 31.1·4-s − 36.1·5-s − 129. i·7-s − 6.88i·8-s + 287. i·10-s + 161.·11-s − 436.·13-s − 1.03e3·14-s − 1.05e3·16-s − 707.·17-s − 2.17e3i·19-s + 1.12e3·20-s − 1.28e3i·22-s + (833. + 2.39e3i)23-s + ⋯
L(s)  = 1  − 1.40i·2-s − 0.972·4-s − 0.646·5-s − 1.00i·7-s − 0.0380i·8-s + 0.908i·10-s + 0.402·11-s − 0.716·13-s − 1.40·14-s − 1.02·16-s − 0.594·17-s − 1.38i·19-s + 0.629·20-s − 0.565i·22-s + (0.328 + 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.581 - 0.813i$
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: \((2,\ 207,\ (\ :5/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2131420115\)
\(L(\frac12)\) \(\approx\) \(0.2131420115\)
\(L(\frac{7}{2})\) 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 \( 1 + (-833. - 2.39e3i)T \)
good2 \( 1 + 7.94iT - 32T^{2} \)
5 \( 1 + 36.1T + 3.12e3T^{2} \)
7 \( 1 + 129. iT - 1.68e4T^{2} \)
11 \( 1 - 161.T + 1.61e5T^{2} \)
13 \( 1 + 436.T + 3.71e5T^{2} \)
17 \( 1 + 707.T + 1.41e6T^{2} \)
19 \( 1 + 2.17e3iT - 2.47e6T^{2} \)
29 \( 1 - 3.68e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.40e3T + 2.86e7T^{2} \)
37 \( 1 + 1.02e4iT - 6.93e7T^{2} \)
41 \( 1 - 7.24e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.46e4iT - 2.29e8T^{2} \)
53 \( 1 - 6.08e3T + 4.18e8T^{2} \)
59 \( 1 + 1.74e3iT - 7.14e8T^{2} \)
61 \( 1 + 3.38e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.62e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.32e4iT - 1.80e9T^{2} \)
73 \( 1 + 841.T + 2.07e9T^{2} \)
79 \( 1 - 6.82e4iT - 3.07e9T^{2} \)
83 \( 1 + 3.78e4T + 3.93e9T^{2} \)
89 \( 1 + 6.99e4T + 5.58e9T^{2} \)
97 \( 1 - 2.49e3iT - 8.58e9T^{2} \)
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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

−11.09222170320435018545954189185, −9.883487636598569297279221825609, −9.121024596614610953357265880355, −7.62390020650217749811020172960, −6.75163366626826298642484985177, −4.72545489408941894253630637359, −3.88507811896785941229240210977, −2.74082180870365350767155897064, −1.22824921349978515930680791313, −0.06791846104156114363201620792, 2.30628745578192646673342423127, 4.11945415168837953685148111362, 5.32348027454482503162340587675, 6.26011784878484303423149435201, 7.25553705339386627878231913820, 8.253110868145123693360382777819, 8.889179376425868537972265839271, 10.19249699485747715072452105212, 11.70615230812457751155611307390, 12.18011418679763940983137547416

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