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Properties

Label 2-833-119.67-c1-0-35
Degree $2$
Conductor $833$
Sign $0.574 + 0.818i$
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  + (0.558 + 0.966i)2-s + (−1.69 − 0.975i)3-s + (0.376 − 0.652i)4-s + (1.29 − 0.749i)5-s − 2.17i·6-s + 3.07·8-s + (0.404 + 0.700i)9-s + (1.44 + 0.836i)10-s + (−0.854 − 0.493i)11-s + (−1.27 + 0.735i)12-s + 5.99·13-s − 2.92·15-s + (0.962 + 1.66i)16-s + (−4.12 − 0.155i)17-s + (−0.451 + 0.781i)18-s + (−2.38 − 4.13i)19-s + ⋯
L(s)  = 1  + (0.394 + 0.683i)2-s + (−0.975 − 0.563i)3-s + (0.188 − 0.326i)4-s + (0.580 − 0.335i)5-s − 0.889i·6-s + 1.08·8-s + (0.134 + 0.233i)9-s + (0.458 + 0.264i)10-s + (−0.257 − 0.148i)11-s + (−0.367 + 0.212i)12-s + 1.66·13-s − 0.755·15-s + (0.240 + 0.416i)16-s + (−0.999 − 0.0378i)17-s + (−0.106 + 0.184i)18-s + (−0.547 − 0.948i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47103 - 0.764229i\)
\(L(\frac12)\) \(\approx\) \(1.47103 - 0.764229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.12 + 0.155i)T \)
good2 \( 1 + (-0.558 - 0.966i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.69 + 0.975i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.29 + 0.749i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.854 + 0.493i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.99T + 13T^{2} \)
19 \( 1 + (2.38 + 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.807 + 0.466i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + (-1.49 - 0.863i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.78 + 3.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.93iT - 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + (1.68 + 2.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.06 + 5.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.33 + 5.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.508 + 0.293i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.414 + 0.717i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 + (-6.04 - 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.0 - 6.93i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.776T + 83T^{2} \)
89 \( 1 + (-2.05 - 3.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−10.27717106630534099691905201721, −9.130463402305738117011021481123, −8.291951456552624225477974663646, −7.00125782219619327217361119497, −6.49182222612513142061087506186, −5.78780699237402195412794877060, −5.18900262774008565473486604423, −4.01696498071171556752773589847, −2.09731829418034522084713968197, −0.851408012030401378719557818789, 1.64044979015583883931899813320, 2.88834439543483075956355722302, 4.04638058138357284045229089763, 4.78076410591545723497900158374, 6.03451100833405478945671243778, 6.45242704142157638394633487254, 7.84829697546058358239009431071, 8.693855875893417992005933616653, 10.04538960735165715063312646311, 10.50366375730109256509075809984

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