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Properties

Label 2-206-103.46-c1-0-7
Degree $2$
Conductor $206$
Sign $0.834 - 0.551i$
Analytic cond. $1.64491$
Root an. cond. $1.28254$
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  + (0.5 + 0.866i)2-s + 2·3-s + (−0.499 + 0.866i)4-s + (1 − 1.73i)5-s + (1 + 1.73i)6-s − 0.999·8-s + 9-s + 1.99·10-s + (−0.999 + 1.73i)12-s − 2·13-s + (2 − 3.46i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + (0.5 + 0.866i)18-s + (−1 − 1.73i)19-s + (0.999 + 1.73i)20-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 1.15·3-s + (−0.249 + 0.433i)4-s + (0.447 − 0.774i)5-s + (0.408 + 0.707i)6-s − 0.353·8-s + 0.333·9-s + 0.632·10-s + (−0.288 + 0.499i)12-s − 0.554·13-s + (0.516 − 0.894i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + (0.117 + 0.204i)18-s + (−0.229 − 0.397i)19-s + (0.223 + 0.387i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 206 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(206\)    =    \(2 \cdot 103\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(1.64491\)
Root analytic conductor: \(1.28254\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{206} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 206,\ (\ :1/2),\ 0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83800 + 0.552285i\)
\(L(\frac12)\) \(\approx\) \(1.83800 + 0.552285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
103 \( 1 + (-10 - 1.73i)T \)
good3 \( 1 - 2T + 3T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + (7 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.94183478188430511938989955808, −11.83703969510683728897979230799, −10.26492441486943179318037494496, −9.116438464983217811635637083347, −8.569330729453508916382359872209, −7.61145799850211050764669888869, −6.28544167734757414228536079061, −5.03506232483960009380372706301, −3.81353668984295986008598968921, −2.25986991265905305405720668491, 2.28114304826638684983775902776, 3.05277376291281144082194601118, 4.47967091578696093696330695786, 6.04674917699901412033587612725, 7.29902487470949888835688159897, 8.500569149411759326659108716548, 9.525372906212732530663740840033, 10.25053732357731402487780095875, 11.35932774178449581880084840330, 12.36278083495067501489694967266

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