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Properties

Label 2-861-7.2-c1-0-51
Degree $2$
Conductor $861$
Sign $-0.682 - 0.731i$
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  + (1.29 − 2.24i)2-s + (−0.5 − 0.866i)3-s + (−2.35 − 4.07i)4-s + (1.72 − 2.98i)5-s − 2.58·6-s + (−0.431 − 2.61i)7-s − 6.99·8-s + (−0.499 + 0.866i)9-s + (−4.46 − 7.74i)10-s + (1.37 + 2.37i)11-s + (−2.35 + 4.07i)12-s + 1.59·13-s + (−6.41 − 2.41i)14-s − 3.45·15-s + (−4.35 + 7.54i)16-s + (3.35 + 5.80i)17-s + ⋯
L(s)  = 1  + (0.915 − 1.58i)2-s + (−0.288 − 0.499i)3-s + (−1.17 − 2.03i)4-s + (0.771 − 1.33i)5-s − 1.05·6-s + (−0.163 − 0.986i)7-s − 2.47·8-s + (−0.166 + 0.288i)9-s + (−1.41 − 2.44i)10-s + (0.413 + 0.715i)11-s + (−0.678 + 1.17i)12-s + 0.443·13-s + (−1.71 − 0.644i)14-s − 0.891·15-s + (−1.08 + 1.88i)16-s + (0.813 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.682 - 0.731i$
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: \((2,\ 861,\ (\ :1/2),\ -0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984316 + 2.26449i\)
\(L(\frac12)\) \(\approx\) \(0.984316 + 2.26449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.431 + 2.61i)T \)
41 \( 1 - T \)
good2 \( 1 + (-1.29 + 2.24i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.37 - 2.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.59T + 13T^{2} \)
17 \( 1 + (-3.35 - 5.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0876 - 0.151i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + (-2.20 - 3.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.21 - 7.30i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 6.34T + 43T^{2} \)
47 \( 1 + (3.31 - 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.17 + 5.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.63 + 2.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.09 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.18 + 5.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + (-7.65 - 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.47 - 2.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.195T + 83T^{2} \)
89 \( 1 + (1.33 - 2.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−10.03338354095348978108626402126, −9.162315171078696394026436676711, −8.196996693939549249845959554391, −6.69143790560100045563056160602, −5.78409614747236938811746039313, −4.82206440739761269478526707603, −4.28314806930716335392245787494, −3.00981485247880694140112544094, −1.46936671614986890618886922350, −1.12326483009500398833303328046, 2.84903560297476133248324337297, 3.49202458799882206431025413538, 4.85638928901280756646308814374, 5.89687867961999676402156252651, 6.00738136269590684638153232289, 6.93841099606390831401019441588, 7.86989687828575876791306087699, 8.905512266442784538395368066492, 9.649163408900997877651909174764, 10.63542323440210526529884498291

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