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Properties

Label 2-833-119.16-c1-0-16
Degree $2$
Conductor $833$
Sign $-0.968 - 0.249i$
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  + (−1.22 + 2.12i)2-s + (1.01 − 0.583i)3-s + (−2.00 − 3.47i)4-s + (2.29 + 1.32i)5-s + 2.86i·6-s + 4.94·8-s + (−0.819 + 1.41i)9-s + (−5.62 + 3.24i)10-s + (−3.03 + 1.75i)11-s + (−4.05 − 2.34i)12-s + 1.85·13-s + 3.09·15-s + (−2.04 + 3.54i)16-s + (1.59 + 3.80i)17-s + (−2.00 − 3.47i)18-s + (0.224 − 0.389i)19-s + ⋯
L(s)  = 1  + (−0.867 + 1.50i)2-s + (0.583 − 0.336i)3-s + (−1.00 − 1.73i)4-s + (1.02 + 0.592i)5-s + 1.16i·6-s + 1.74·8-s + (−0.273 + 0.472i)9-s + (−1.77 + 1.02i)10-s + (−0.915 + 0.528i)11-s + (−1.17 − 0.676i)12-s + 0.513·13-s + 0.798·15-s + (−0.511 + 0.885i)16-s + (0.387 + 0.921i)17-s + (−0.473 − 0.820i)18-s + (0.0515 − 0.0893i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.968 - 0.249i$
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: \((2,\ 833,\ (\ :1/2),\ -0.968 - 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140806 + 1.11145i\)
\(L(\frac12)\) \(\approx\) \(0.140806 + 1.11145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 3.80i)T \)
good2 \( 1 + (1.22 - 2.12i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.01 + 0.583i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.29 - 1.32i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.03 - 1.75i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
19 \( 1 + (-0.224 + 0.389i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.18 + 2.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.72iT - 29T^{2} \)
31 \( 1 + (-5.78 + 3.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.47 - 2.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.10iT - 41T^{2} \)
43 \( 1 + 7.42T + 43T^{2} \)
47 \( 1 + (6.16 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.15 - 8.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.29 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.27 + 1.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.03 - 1.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (1.95 - 1.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.384 - 0.222i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.97T + 83T^{2} \)
89 \( 1 + (0.598 - 1.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.92iT - 97T^{2} \)
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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

−10.25976411099804227123966889105, −9.593851314693043445639324072416, −8.581418842769608506942228582413, −8.049750464979920104516144606584, −7.32324747378835406357169614396, −6.35212969998112927627036482801, −5.81203514352400168965344519685, −4.79821869181507408143423135345, −2.89028238720390634914543786873, −1.69425543117795749690652771006, 0.68429425803735148937868929958, 2.06185455331027516449550971897, 2.94366999547392587569030739118, 3.84896386343126094848873796920, 5.19710156390600463037232169639, 6.22347900447406404253488826295, 7.962894388375689185403270616931, 8.453866928849739138619978395233, 9.273544525024706444598845558749, 9.906051542980499876319182456818

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