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Properties

Label 2-833-119.16-c1-0-53
Degree $2$
Conductor $833$
Sign $-0.910 - 0.412i$
Analytic cond. $6.65153$
Root an. cond. $2.57905$
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  + (1.27 − 2.20i)2-s + (1.86 − 1.07i)3-s + (−2.24 − 3.89i)4-s + (−1.61 − 0.932i)5-s − 5.49i·6-s − 6.36·8-s + (0.824 − 1.42i)9-s + (−4.11 + 2.37i)10-s + (3.37 − 1.94i)11-s + (−8.39 − 4.84i)12-s − 4.67·13-s − 4.02·15-s + (−3.60 + 6.25i)16-s + (3.94 + 1.21i)17-s + (−2.10 − 3.64i)18-s + (−1.25 + 2.17i)19-s + ⋯
L(s)  = 1  + (0.901 − 1.56i)2-s + (1.07 − 0.622i)3-s + (−1.12 − 1.94i)4-s + (−0.722 − 0.416i)5-s − 2.24i·6-s − 2.24·8-s + (0.274 − 0.476i)9-s + (−1.30 + 0.751i)10-s + (1.01 − 0.587i)11-s + (−2.42 − 1.39i)12-s − 1.29·13-s − 1.03·15-s + (−0.902 + 1.56i)16-s + (0.955 + 0.294i)17-s + (−0.495 − 0.858i)18-s + (−0.288 + 0.499i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.910 - 0.412i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592535 + 2.74313i\)
\(L(\frac12)\) \(\approx\) \(0.592535 + 2.74313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 + (-3.94 - 1.21i)T \)
good2 \( 1 + (-1.27 + 2.20i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.86 + 1.07i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.61 + 0.932i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.37 + 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
19 \( 1 + (1.25 - 2.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.14 - 3.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.19iT - 29T^{2} \)
31 \( 1 + (-4.16 + 2.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.77 + 4.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.670iT - 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + (4.10 - 7.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.18 - 2.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.86 + 6.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.580 - 0.335i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.92 + 3.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + (-2.14 + 1.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.67T + 83T^{2} \)
89 \( 1 + (-3.65 + 6.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.76iT - 97T^{2} \)
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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

−9.723991652484115521026697553274, −9.150897739381929278703317990073, −8.154674113015517008070284378026, −7.36983100833371353818518610453, −5.93229648649219355197603741822, −4.80695627214672803709747983877, −3.86533626155468735662799966108, −3.14229021661039755110468812958, −2.14158176125355045878335197474, −0.976848550914503715515898974537, 2.86982030358680253571648983918, 3.62259881705329930645601428780, 4.50143654884081990150408076621, 5.18453125555723887714998137239, 6.67686453355446496245373537121, 7.15343494942091122149695328937, 7.907218533063350226229597734524, 8.823472173312969950001956313688, 9.380169654735654763878505163481, 10.48648282476938680153829223470

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