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Properties

Label 2-833-119.27-c1-0-0
Degree $2$
Conductor $833$
Sign $-0.397 + 0.917i$
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  + (−0.684 + 1.65i)2-s + (0.302 − 0.453i)3-s + (−0.846 − 0.846i)4-s + (0.523 + 2.62i)5-s + (0.541 + 0.810i)6-s + (−1.32 + 0.549i)8-s + (1.03 + 2.49i)9-s + (−4.70 − 0.935i)10-s + (−2.78 − 4.16i)11-s + (−0.640 + 0.127i)12-s + (−4.70 − 4.70i)13-s + (1.35 + 0.559i)15-s − 4.96i·16-s + (−2.09 + 3.55i)17-s − 4.83·18-s + (−4.91 − 2.03i)19-s + ⋯
L(s)  = 1  + (−0.483 + 1.16i)2-s + (0.174 − 0.261i)3-s + (−0.423 − 0.423i)4-s + (0.233 + 1.17i)5-s + (0.221 + 0.330i)6-s + (−0.468 + 0.194i)8-s + (0.344 + 0.832i)9-s + (−1.48 − 0.295i)10-s + (−0.839 − 1.25i)11-s + (−0.184 + 0.0367i)12-s + (−1.30 − 1.30i)13-s + (0.348 + 0.144i)15-s − 1.24i·16-s + (−0.507 + 0.861i)17-s − 1.13·18-s + (−1.12 − 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220170 - 0.335458i\)
\(L(\frac12)\) \(\approx\) \(0.220170 - 0.335458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 3.55i)T \)
good2 \( 1 + (0.684 - 1.65i)T + (-1.41 - 1.41i)T^{2} \)
3 \( 1 + (-0.302 + 0.453i)T + (-1.14 - 2.77i)T^{2} \)
5 \( 1 + (-0.523 - 2.62i)T + (-4.61 + 1.91i)T^{2} \)
11 \( 1 + (2.78 + 4.16i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (4.70 + 4.70i)T + 13iT^{2} \)
19 \( 1 + (4.91 + 2.03i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.40 - 2.27i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.90 - 9.55i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (0.638 + 0.426i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-4.93 - 3.29i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.0510 + 0.256i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-0.397 - 0.959i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (6.07 + 6.07i)T + 47iT^{2} \)
53 \( 1 + (0.700 - 1.69i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.28 + 5.51i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.48 - 0.295i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + (3.95 + 2.64i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (0.221 + 1.11i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (6.72 + 10.0i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (5.11 - 12.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.38 - 1.38i)T - 89iT^{2} \)
97 \( 1 + (9.84 - 1.95i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.54974233580781823455298901856, −10.06123938822541176260773935797, −8.640239510246239827680521646834, −8.094191399421462507325775965173, −7.40320210935821991788015715168, −6.66822048543443669511571191033, −5.83706346800781741692308478102, −4.98378869529086967775921781411, −3.12476053482283953205646029316, −2.42089633041722813563040826390, 0.20395957660603844485835396510, 1.83717331662688224511383701116, 2.54097298965505332408004205729, 4.36582716510658968825443274013, 4.53546500809715043950048720916, 6.11961611205727178571142944915, 7.14928491378444960933829133763, 8.335423845070692240296509235352, 9.249456236813812713691579486174, 9.693812738205096963313464091037

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