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Properties

Label 2-861-7.4-c1-0-43
Degree $2$
Conductor $861$
Sign $-0.991 + 0.126i$
Analytic cond. $6.87511$
Root an. cond. $2.62204$
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  + (−0.406 − 0.703i)2-s + (0.5 − 0.866i)3-s + (0.669 − 1.16i)4-s + (0.5 + 0.866i)5-s − 0.812·6-s + (−2 + 1.73i)7-s − 2.71·8-s + (−0.499 − 0.866i)9-s + (0.406 − 0.703i)10-s + (2.57 − 4.46i)11-s + (−0.669 − 1.16i)12-s − 2.18·13-s + (2.03 + 0.703i)14-s + 0.999·15-s + (−0.236 − 0.409i)16-s + (−2.93 + 5.08i)17-s + ⋯
L(s)  = 1  + (−0.287 − 0.497i)2-s + (0.288 − 0.499i)3-s + (0.334 − 0.580i)4-s + (0.223 + 0.387i)5-s − 0.331·6-s + (−0.755 + 0.654i)7-s − 0.959·8-s + (−0.166 − 0.288i)9-s + (0.128 − 0.222i)10-s + (0.776 − 1.34i)11-s + (−0.193 − 0.334i)12-s − 0.606·13-s + (0.543 + 0.188i)14-s + 0.258·15-s + (−0.0591 − 0.102i)16-s + (−0.711 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0628440 - 0.990290i\)
\(L(\frac12)\) \(\approx\) \(0.0628440 - 0.990290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
41 \( 1 + T \)
good2 \( 1 + (0.406 + 0.703i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.57 + 4.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.66 + 4.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.74 + 6.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + (-2.38 + 4.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.98 + 8.62i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 9.05T + 43T^{2} \)
47 \( 1 + (-4.24 - 7.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.76 - 6.51i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.482 - 0.835i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.45 - 4.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.23 + 2.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + (-1.83 + 3.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.07 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-1.88 - 3.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.87T + 97T^{2} \)
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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

−9.711110760606337230074168464824, −8.969809963658160276380654698567, −8.432041995191139723390815681838, −6.92035643996337715994921878997, −6.24164900134488944672924661352, −5.83993677999014636135417979726, −4.05733099010096233197026876748, −2.73950644149685218163065390433, −2.18336961495809379800273945091, −0.46604757171921009104179855475, 2.00110066687348543757546330172, 3.35485759861785805702688971177, 4.18326965073349237018898408754, 5.28638082863310805168147617757, 6.61336986483172075204634686350, 7.13371853267342200336533146581, 7.937009308212934004355887782644, 9.073451529759784118749981192112, 9.547774657864527772314975629482, 10.20998365085161271585549612036

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