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Properties

Label 2-207-23.16-c1-0-0
Degree $2$
Conductor $207$
Sign $-0.547 - 0.836i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.460 + 0.531i)2-s + (0.214 + 1.48i)4-s + (0.700 + 1.53i)5-s + (−3.11 + 0.915i)7-s + (−2.07 − 1.33i)8-s + (−1.13 − 0.334i)10-s + (1.33 + 1.54i)11-s + (−0.227 − 0.0666i)13-s + (0.949 − 2.07i)14-s + (−1.22 + 0.358i)16-s + (0.427 − 2.97i)17-s + (0.682 + 4.74i)19-s + (−2.13 + 1.37i)20-s − 1.43·22-s + (0.779 + 4.73i)23-s + ⋯
L(s)  = 1  + (−0.325 + 0.376i)2-s + (0.107 + 0.744i)4-s + (0.313 + 0.685i)5-s + (−1.17 + 0.345i)7-s + (−0.733 − 0.471i)8-s + (−0.360 − 0.105i)10-s + (0.403 + 0.465i)11-s + (−0.0629 − 0.0184i)13-s + (0.253 − 0.555i)14-s + (−0.305 + 0.0897i)16-s + (0.103 − 0.721i)17-s + (0.156 + 1.08i)19-s + (−0.477 + 0.306i)20-s − 0.306·22-s + (0.162 + 0.986i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.547 - 0.836i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ -0.547 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424080 + 0.784687i\)
\(L(\frac12)\) \(\approx\) \(0.424080 + 0.784687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.779 - 4.73i)T \)
good2 \( 1 + (0.460 - 0.531i)T + (-0.284 - 1.97i)T^{2} \)
5 \( 1 + (-0.700 - 1.53i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (3.11 - 0.915i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-1.33 - 1.54i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (0.227 + 0.0666i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.427 + 2.97i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.682 - 4.74i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (1.01 - 7.03i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-5.64 - 3.62i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-4.77 + 10.4i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (3.40 + 7.46i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.42 + 1.55i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + (-0.510 + 0.149i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (3.30 + 0.971i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-4.08 - 2.62i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (9.77 - 11.2i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-8.80 + 10.1i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.632 + 4.39i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (2.00 + 0.589i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (2.67 - 5.86i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (2.11 - 1.35i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-0.753 - 1.65i)T + (-63.5 + 73.3i)T^{2} \)
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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

−12.49833260216988583752861733880, −12.06144775913955639390355804489, −10.63220330578544055455210836994, −9.594008720239245577533623039621, −8.896886188178075570864562715734, −7.44813359188337364741012603051, −6.80661086123276653207419321500, −5.75749326421248847963982115448, −3.75216381389828221287595911680, −2.70852926813595541028880874568, 0.874909118337202849955886946712, 2.78018530873677045664223260998, 4.52186597454481691393435043054, 5.94394358370258843144431712034, 6.64623182984643761924942426779, 8.387679298713408066047084523054, 9.380211713175968390196379927697, 9.945262069557303373510050600676, 10.97693927998943804065782128663, 11.97184603036704700982578974070

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