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Properties

Label 2-861-21.5-c1-0-80
Degree $2$
Conductor $861$
Sign $-0.132 - 0.991i$
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  + (2.15 + 1.24i)2-s + (0.624 + 1.61i)3-s + (2.09 + 3.62i)4-s + (1.86 − 3.23i)5-s + (−0.664 + 4.25i)6-s + (1.06 + 2.42i)7-s + 5.44i·8-s + (−2.22 + 2.01i)9-s + (8.05 − 4.65i)10-s + (1.36 − 0.785i)11-s + (−4.55 + 5.64i)12-s − 2.25i·13-s + (−0.718 + 6.54i)14-s + (6.39 + 0.998i)15-s + (−2.58 + 4.47i)16-s + (−3.44 − 5.97i)17-s + ⋯
L(s)  = 1  + (1.52 + 0.879i)2-s + (0.360 + 0.932i)3-s + (1.04 + 1.81i)4-s + (0.836 − 1.44i)5-s + (−0.271 + 1.73i)6-s + (0.402 + 0.915i)7-s + 1.92i·8-s + (−0.740 + 0.672i)9-s + (2.54 − 1.47i)10-s + (0.410 − 0.236i)11-s + (−1.31 + 1.63i)12-s − 0.625i·13-s + (−0.191 + 1.74i)14-s + (1.65 + 0.257i)15-s + (−0.646 + 1.11i)16-s + (−0.836 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.132 - 0.991i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.132 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01165 + 3.44063i\)
\(L(\frac12)\) \(\approx\) \(3.01165 + 3.44063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.624 - 1.61i)T \)
7 \( 1 + (-1.06 - 2.42i)T \)
41 \( 1 + T \)
good2 \( 1 + (-2.15 - 1.24i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.86 + 3.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.36 + 0.785i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.25iT - 13T^{2} \)
17 \( 1 + (3.44 + 5.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.08 - 0.629i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.97 + 3.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 + (6.09 - 3.52i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.43 - 5.95i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + (-4.01 + 6.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.90 - 2.25i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.56 - 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.77 - 2.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.40 - 5.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.12iT - 71T^{2} \)
73 \( 1 + (-2.63 + 1.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.18 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (3.12 - 5.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.03iT - 97T^{2} \)
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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

−10.33212533736580215224790950083, −9.175157538087413348552743318383, −8.760319237237643025608669489477, −7.921678560694089128633904910006, −6.54671974710123922061192139730, −5.47927218985402190223418345273, −5.18603479572758287408718878807, −4.53474115518686229211867007494, −3.35222626239975167134499356796, −2.18248359546577471006172272230, 1.84592400069562337878523967661, 2.12795792368687837739512450840, 3.53607124269515115700773135869, 4.08334306873835678428161793908, 5.68078848973302157297916479214, 6.37030628655416475531335857603, 6.92869399539921319977488219651, 7.939225792895064460491719797969, 9.502389194020568390221863789939, 10.27864786506115780600713840603

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