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Properties

Label 2-833-119.16-c1-0-40
Degree $2$
Conductor $833$
Sign $0.979 + 0.199i$
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  + (−0.717 + 1.24i)2-s + (2.53 − 1.46i)3-s + (−0.0287 − 0.0498i)4-s + (−1.51 − 0.874i)5-s + 4.19i·6-s − 2.78·8-s + (2.77 − 4.80i)9-s + (2.17 − 1.25i)10-s + (1.21 − 0.703i)11-s + (−0.145 − 0.0841i)12-s + 2.32·13-s − 5.11·15-s + (2.05 − 3.56i)16-s + (−1.79 − 3.71i)17-s + (3.97 + 6.88i)18-s + (3.72 − 6.44i)19-s + ⋯
L(s)  = 1  + (−0.507 + 0.878i)2-s + (1.46 − 0.843i)3-s + (−0.0143 − 0.0249i)4-s + (−0.677 − 0.390i)5-s + 1.71i·6-s − 0.985·8-s + (0.924 − 1.60i)9-s + (0.686 − 0.396i)10-s + (0.367 − 0.212i)11-s + (−0.0420 − 0.0242i)12-s + 0.644·13-s − 1.31·15-s + (0.513 − 0.890i)16-s + (−0.434 − 0.900i)17-s + (0.937 + 1.62i)18-s + (0.854 − 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.979 + 0.199i$
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: \((2,\ 833,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80638 - 0.181926i\)
\(L(\frac12)\) \(\approx\) \(1.80638 - 0.181926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 3.71i)T \)
good2 \( 1 + (0.717 - 1.24i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.53 + 1.46i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.51 + 0.874i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.21 + 0.703i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.32T + 13T^{2} \)
19 \( 1 + (-3.72 + 6.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.60 - 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.93iT - 29T^{2} \)
31 \( 1 + (-4.64 + 2.68i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.55 - 0.898i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.04iT - 41T^{2} \)
43 \( 1 + 1.00T + 43T^{2} \)
47 \( 1 + (1.93 - 3.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.90 - 5.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.01 + 5.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.56 + 4.36i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.08 + 1.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.93iT - 71T^{2} \)
73 \( 1 + (-5.83 + 3.36i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.68 + 0.975i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + (-0.454 + 0.787i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−9.366277280544673219857822070771, −9.117217545892846025773861295735, −8.361663615282329495422781281749, −7.67881525990916777916855555247, −7.09153065322700764795741752039, −6.37832729406321951349165498166, −4.82278384181601666145994024896, −3.40048011317819535981198899124, −2.76158226046952567326762668711, −0.983815129628827058408747480407, 1.57768871543713154015546427363, 2.78237008639651634668772840053, 3.56343209112499152950720987383, 4.23516673185865862494689569494, 5.82329855502769872589080210276, 7.07773078780570192505063290300, 8.223916526053328311075913224945, 8.626114488503608089041832617060, 9.552102812031906049777469609185, 10.13868849254843216529836170853

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