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Properties

Label 2-861-41.16-c1-0-11
Degree $2$
Conductor $861$
Sign $-0.158 - 0.987i$
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  + (−0.0301 + 0.0929i)2-s − 3-s + (1.61 + 1.16i)4-s + (1.86 + 1.35i)5-s + (0.0301 − 0.0929i)6-s + (0.309 + 0.951i)7-s + (−0.315 + 0.229i)8-s + 9-s + (−0.182 + 0.132i)10-s + (−1.97 + 1.43i)11-s + (−1.61 − 1.16i)12-s + (−1.29 + 3.98i)13-s − 0.0976·14-s + (−1.86 − 1.35i)15-s + (1.21 + 3.74i)16-s + (3.24 − 2.35i)17-s + ⋯
L(s)  = 1  + (−0.0213 + 0.0656i)2-s − 0.577·3-s + (0.805 + 0.584i)4-s + (0.834 + 0.606i)5-s + (0.0123 − 0.0379i)6-s + (0.116 + 0.359i)7-s + (−0.111 + 0.0810i)8-s + 0.333·9-s + (−0.0576 + 0.0418i)10-s + (−0.595 + 0.432i)11-s + (−0.464 − 0.337i)12-s + (−0.359 + 1.10i)13-s − 0.0261·14-s + (−0.481 − 0.350i)15-s + (0.304 + 0.937i)16-s + (0.786 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.158 - 0.987i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.158 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06734 + 1.25216i\)
\(L(\frac12)\) \(\approx\) \(1.06734 + 1.25216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-5.38 + 3.46i)T \)
good2 \( 1 + (0.0301 - 0.0929i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.86 - 1.35i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (1.97 - 1.43i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.29 - 3.98i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.24 + 2.35i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0538 + 0.165i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.477 - 1.47i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.88 + 3.55i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.45 - 1.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.09 + 1.52i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (1.22 - 3.76i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.32 + 10.2i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.54 - 4.02i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.09 - 6.44i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.00 - 9.25i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.16 - 5.93i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (7.46 - 5.42i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 9.08T + 83T^{2} \)
89 \( 1 + (3.16 + 9.75i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.50 - 1.82i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.37583327827860549831515959240, −9.783721923033565168324484264374, −8.753571668574021474776764649168, −7.45127951919127602839804970186, −7.08854527910763476914736685543, −6.05088514050473181883638528247, −5.40631771380718767429490055339, −4.05571515393434222000386492607, −2.66709844423024153918851430469, −1.92059361150704505076600260139, 0.843964046488592495299238010190, 1.99581169932531308267209061233, 3.34101138130535960621652969690, 5.00470588294084127987449329158, 5.57618326674704675054021450109, 6.19263574312683722281031405194, 7.34402363049472070427131361273, 8.102118353062588344385177085562, 9.404350343105834065382146112961, 10.10278063075916994187007431385

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