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Properties

Label 2-858-13.3-c1-0-5
Degree $2$
Conductor $858$
Sign $-0.859 - 0.511i$
Analytic cond. $6.85116$
Root an. cond. $2.61747$
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  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (−0.499 + 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s − 0.999·12-s + (−1 + 3.46i)13-s + 1.99·14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.204 + 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.150 + 0.261i)11-s − 0.288·12-s + (−0.277 + 0.960i)13-s + 0.534·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(858\)    =    \(2 \cdot 3 \cdot 11 \cdot 13\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(6.85116\)
Root analytic conductor: \(2.61747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{858} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 858,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405601 + 1.47581i\)
\(L(\frac12)\) \(\approx\) \(0.405601 + 1.47581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (1 - 3.46i)T \)
good5 \( 1 + T + 5T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.54298248671324550038453604748, −9.534565387747137288961807639639, −8.749746893915327223239997630009, −7.86931391100553822103305836229, −7.20594586350111001523392886684, −6.21946148150173000014127390597, −5.09961086730724949522335620101, −4.10578347140989261502493534280, −3.70150179719989405192888694377, −1.91116599211117776189249755055, 0.63333450127094566975178159467, 2.35596969912902314816484243583, 2.99943053835457662284924266747, 4.41072817172552002562531423737, 5.19916201482602066574246735125, 6.34576195383975651338000875377, 7.24054507725785735858594045631, 8.382081225215226294899857127661, 8.810633868796006251179546082571, 9.896988220614733028413819635255

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