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Properties

Label 2-861-861.524-c1-0-11
Degree $2$
Conductor $861$
Sign $-0.802 - 0.597i$
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  − 2.09·2-s + (0.839 + 1.51i)3-s + 2.40·4-s + 2.02i·5-s + (−1.76 − 3.17i)6-s + (2.04 − 1.67i)7-s − 0.839·8-s + (−1.58 + 2.54i)9-s − 4.24i·10-s + (−1.97 − 1.97i)11-s + (2.01 + 3.63i)12-s + (−0.746 − 0.746i)13-s + (−4.29 + 3.51i)14-s + (−3.06 + 1.70i)15-s − 3.03·16-s + (4.42 + 4.42i)17-s + ⋯
L(s)  = 1  − 1.48·2-s + (0.484 + 0.874i)3-s + 1.20·4-s + 0.905i·5-s + (−0.719 − 1.29i)6-s + (0.774 − 0.632i)7-s − 0.296·8-s + (−0.529 + 0.848i)9-s − 1.34i·10-s + (−0.596 − 0.596i)11-s + (0.581 + 1.04i)12-s + (−0.207 − 0.207i)13-s + (−1.14 + 0.938i)14-s + (−0.791 + 0.438i)15-s − 0.759·16-s + (1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.802 - 0.597i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.802 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221398 + 0.668176i\)
\(L(\frac12)\) \(\approx\) \(0.221398 + 0.668176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.839 - 1.51i)T \)
7 \( 1 + (-2.04 + 1.67i)T \)
41 \( 1 + (-0.494 - 6.38i)T \)
good2 \( 1 + 2.09T + 2T^{2} \)
5 \( 1 - 2.02iT - 5T^{2} \)
11 \( 1 + (1.97 + 1.97i)T + 11iT^{2} \)
13 \( 1 + (0.746 + 0.746i)T + 13iT^{2} \)
17 \( 1 + (-4.42 - 4.42i)T + 17iT^{2} \)
19 \( 1 + (3.72 - 3.72i)T - 19iT^{2} \)
23 \( 1 - 1.99iT - 23T^{2} \)
29 \( 1 + (2.41 + 2.41i)T + 29iT^{2} \)
31 \( 1 - 6.69iT - 31T^{2} \)
37 \( 1 + 0.270T + 37T^{2} \)
43 \( 1 - 9.03iT - 43T^{2} \)
47 \( 1 + (3.64 + 3.64i)T + 47iT^{2} \)
53 \( 1 + (2.30 + 2.30i)T + 53iT^{2} \)
59 \( 1 - 2.66T + 59T^{2} \)
61 \( 1 - 0.289T + 61T^{2} \)
67 \( 1 + (5.92 + 5.92i)T + 67iT^{2} \)
71 \( 1 + (-6.99 - 6.99i)T + 71iT^{2} \)
73 \( 1 - 0.182T + 73T^{2} \)
79 \( 1 + (10.5 - 10.5i)T - 79iT^{2} \)
83 \( 1 - 5.98T + 83T^{2} \)
89 \( 1 + (3.63 - 3.63i)T - 89iT^{2} \)
97 \( 1 + (-4.74 + 4.74i)T - 97iT^{2} \)
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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

−10.20873178529386698515937070109, −9.987380511576145844886849275070, −8.721482265705032744008978559605, −8.024681638800742906685946937293, −7.70671657473187177372954019254, −6.46266767134757405542205436369, −5.22300733138267149298814136861, −3.97743091959785450733711982599, −2.93427236846430875699903013021, −1.60023997644837528286285471881, 0.53287114243960105981206792784, 1.77149373554233367188168878563, 2.59148480519590876384910108808, 4.57578647574134679582362491053, 5.52663040012167561048803041965, 6.91926491217208690787333727606, 7.61084124981889595368037987753, 8.227274650481059696331327262669, 9.003706955219108495224941759081, 9.345709623466441763609184277026

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