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Properties

Label 2-861-21.20-c1-0-105
Degree $2$
Conductor $861$
Sign $-0.537 - 0.843i$
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  − 1.93i·2-s + (1 − 1.41i)3-s − 1.73·4-s − 1.73·5-s + (−2.73 − 1.93i)6-s + (1 − 2.44i)7-s − 0.517i·8-s + (−1.00 − 2.82i)9-s + 3.34i·10-s + 3.86i·11-s + (−1.73 + 2.44i)12-s + 1.55i·13-s + (−4.73 − 1.93i)14-s + (−1.73 + 2.44i)15-s − 4.46·16-s − 4.73·17-s + ⋯
L(s)  = 1  − 1.36i·2-s + (0.577 − 0.816i)3-s − 0.866·4-s − 0.774·5-s + (−1.11 − 0.788i)6-s + (0.377 − 0.925i)7-s − 0.183i·8-s + (−0.333 − 0.942i)9-s + 1.05i·10-s + 1.16i·11-s + (−0.500 + 0.707i)12-s + 0.430i·13-s + (−1.26 − 0.516i)14-s + (−0.447 + 0.632i)15-s − 1.11·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612066 + 1.11629i\)
\(L(\frac12)\) \(\approx\) \(0.612066 + 1.11629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
7 \( 1 + (-1 + 2.44i)T \)
41 \( 1 - T \)
good2 \( 1 + 1.93iT - 2T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 3.86iT - 11T^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + 0.138iT - 23T^{2} \)
29 \( 1 + 0.138iT - 29T^{2} \)
31 \( 1 + 9.14iT - 31T^{2} \)
37 \( 1 - 7.19T + 37T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 - 0.464T + 47T^{2} \)
53 \( 1 - 5.41iT - 53T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 - 0.656iT - 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 9.46T + 89T^{2} \)
97 \( 1 + 7.58iT - 97T^{2} \)
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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

−9.572529699436323567706450334990, −9.079916359791174545487671744195, −7.76625809763455871481342649618, −7.32017865452294005018704529008, −6.42719066148462166102232649899, −4.28894176811055848303542505285, −4.19378474326461124031652490969, −2.71441690428333021148086127483, −1.88835432939180709980329852099, −0.55719760954197253478491284762, 2.45589208423413016856202190889, 3.65567863206731390112608476285, 4.70758608878211198057480942904, 5.55378732735667420646966018848, 6.29575163902840905456715315581, 7.60916781416913816968607753956, 8.211105141271339087037532817608, 8.677036851479254301289747382660, 9.439792527287442399135105710279, 10.84861382734559129994831108890

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