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Properties

Label 2-207-69.68-c5-0-3
Degree $2$
Conductor $207$
Sign $0.940 - 0.339i$
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  − 6.22i·2-s − 6.78·4-s − 105.·5-s − 86.6i·7-s − 157. i·8-s + 659. i·10-s − 110.·11-s − 288.·13-s − 539.·14-s − 1.19e3·16-s − 432.·17-s + 503. i·19-s + 718.·20-s + 687. i·22-s + (−2.08e3 − 1.45e3i)23-s + ⋯
L(s)  = 1  − 1.10i·2-s − 0.211·4-s − 1.89·5-s − 0.668i·7-s − 0.867i·8-s + 2.08i·10-s − 0.274·11-s − 0.473·13-s − 0.735·14-s − 1.16·16-s − 0.362·17-s + 0.319i·19-s + 0.401·20-s + 0.302i·22-s + (−0.820 − 0.571i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.940 - 0.339i$
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: \((2,\ 207,\ (\ :5/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3841771626\)
\(L(\frac12)\) \(\approx\) \(0.3841771626\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.08e3 + 1.45e3i)T \)
good2 \( 1 + 6.22iT - 32T^{2} \)
5 \( 1 + 105.T + 3.12e3T^{2} \)
7 \( 1 + 86.6iT - 1.68e4T^{2} \)
11 \( 1 + 110.T + 1.61e5T^{2} \)
13 \( 1 + 288.T + 3.71e5T^{2} \)
17 \( 1 + 432.T + 1.41e6T^{2} \)
19 \( 1 - 503. iT - 2.47e6T^{2} \)
29 \( 1 - 3.39e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.19e3T + 2.86e7T^{2} \)
37 \( 1 - 6.31e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.35e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.42e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.20e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.60e4T + 4.18e8T^{2} \)
59 \( 1 + 3.00e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.37e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.99e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.91e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.28e4T + 2.07e9T^{2} \)
79 \( 1 - 1.28e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.00e5T + 3.93e9T^{2} \)
89 \( 1 + 6.59e4T + 5.58e9T^{2} \)
97 \( 1 + 1.27e5iT - 8.58e9T^{2} \)
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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

−11.63573089099781077722528984627, −10.77535246379865205988353159819, −10.05568035838428356325475893020, −8.545179799474574945844661377286, −7.59203106464740683278379413479, −6.73420577680759619248432164773, −4.56590633294736123166204793108, −3.83652372919257408167295597707, −2.74978590799708117554626744306, −0.941695585492235441746986803185, 0.14615753735100633175821095831, 2.59772270929847720504700064134, 4.12522656677455091975162844674, 5.21474149657918436281985530617, 6.51252189203315054522980554902, 7.54162230574368439990659467752, 8.079734845166537913131702004049, 9.001131533209039666712565402340, 10.66433741013977705124443144112, 11.79394072896009299598954810887

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