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Properties

Label 2-861-7.2-c1-0-2
Degree $2$
Conductor $861$
Sign $-0.930 + 0.365i$
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  + (0.5 + 0.866i)3-s + (1 + 1.73i)4-s + (−1.27 + 2.21i)5-s + (−2.58 − 0.557i)7-s + (−0.499 + 0.866i)9-s + (−3.08 − 5.34i)11-s + (−0.999 + 1.73i)12-s − 2.06·13-s − 2.55·15-s + (−1.99 + 3.46i)16-s + (1 + 1.73i)17-s + (0.223 − 0.387i)19-s − 5.10·20-s + (−0.810 − 2.51i)21-s + (0.276 − 0.478i)23-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.5 + 0.866i)4-s + (−0.570 + 0.988i)5-s + (−0.977 − 0.210i)7-s + (−0.166 + 0.288i)9-s + (−0.930 − 1.61i)11-s + (−0.288 + 0.499i)12-s − 0.573·13-s − 0.659·15-s + (−0.499 + 0.866i)16-s + (0.242 + 0.420i)17-s + (0.0513 − 0.0889i)19-s − 1.14·20-s + (−0.176 − 0.549i)21-s + (0.0575 − 0.0997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113545 - 0.599217i\)
\(L(\frac12)\) \(\approx\) \(0.113545 - 0.599217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.557i)T \)
41 \( 1 + T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.27 - 2.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.08 + 5.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.06T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.223 + 0.387i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.276 + 0.478i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.10T + 29T^{2} \)
31 \( 1 + (-3.86 - 6.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + (-4.15 + 7.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.06 - 3.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.06 + 7.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.87 - 6.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0338 - 0.0585i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.13T + 71T^{2} \)
73 \( 1 + (-5.31 - 9.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.39 - 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + (6.15 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−10.69081476331115395541212064811, −10.01193484219829211232624578265, −8.769590710550955035297395375343, −8.100736962189396230938994952512, −7.24800684450683995963243409825, −6.55239140643983952990202182807, −5.44849364329614145303879181172, −3.87171521844227780051834535057, −3.22530981891750119989669103766, −2.69867613248757695729624363587, 0.25928849703570603498450021626, 1.84968371553826701174054429451, 2.86702189231908063230868945898, 4.46423796734062632391566853913, 5.24301923428664390608309679583, 6.26190652506586346646729841884, 7.27462654846059593715739391250, 7.75131467378251837107452803822, 9.009978576623732823278377762245, 9.752187229112758938475558271074

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