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Properties

Label 2-858-13.3-c1-0-20
Degree $2$
Conductor $858$
Sign $-0.999 + 0.0256i$
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 + (−3.5 + 6.06i)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 + (−0.848 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(858\)    =    \(2 \cdot 3 \cdot 11 \cdot 13\)
Sign: $-0.999 + 0.0256i$
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: \((2,\ 858,\ (\ :1/2),\ -0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00705902 - 0.550437i\)
\(L(\frac12)\) \(\approx\) \(0.00705902 - 0.550437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.945838170999214592142380828289, −8.853314205065558989377154578293, −8.308395424856630153098979570794, −7.24034130576638999181566814403, −6.17646916613760798657120305128, −5.52074374943220399817004471947, −4.21408026986183661387275376243, −2.81419295998712926770135363591, −2.02892993319840375567882470909, −0.29933782017153262909137973045, 1.67502167164873065267911796920, 3.41752078538555404266606607512, 4.52358791548157346999096809119, 5.38974303259730280145641286933, 6.33376794784075563692210346435, 7.14979237725456585024624539404, 7.909797151392838106110619348715, 9.203917846168703148779175071667, 9.725932128143922681623133968296, 10.16798144205014045999269306267

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