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Properties

Label 2-833-17.9-c1-0-20
Degree $2$
Conductor $833$
Sign $-0.758 - 0.651i$
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  + (−0.292 + 0.292i)2-s + (2.41 + i)3-s + 1.82i·4-s + (−0.707 + 1.70i)5-s + (−1 + 0.414i)6-s + (−1.12 − 1.12i)8-s + (2.70 + 2.70i)9-s + (−0.292 − 0.707i)10-s + (−1 + 0.414i)11-s + (−1.82 + 4.41i)12-s + 1.41i·13-s + (−3.41 + 3.41i)15-s − 3·16-s + (2.82 − 3i)17-s − 1.58·18-s + (−3.41 + 3.41i)19-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (1.39 + 0.577i)3-s + 0.914i·4-s + (−0.316 + 0.763i)5-s + (−0.408 + 0.169i)6-s + (−0.396 − 0.396i)8-s + (0.902 + 0.902i)9-s + (−0.0926 − 0.223i)10-s + (−0.301 + 0.124i)11-s + (−0.527 + 1.27i)12-s + 0.392i·13-s + (−0.881 + 0.881i)15-s − 0.750·16-s + (0.685 − 0.727i)17-s − 0.373·18-s + (−0.783 + 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.758 - 0.651i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665206 + 1.79598i\)
\(L(\frac12)\) \(\approx\) \(0.665206 + 1.79598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 + 3i)T \)
good2 \( 1 + (0.292 - 0.292i)T - 2iT^{2} \)
3 \( 1 + (-2.41 - i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.707 - 1.70i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (1 - 0.414i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (3.41 - 3.41i)T - 19iT^{2} \)
23 \( 1 + (-3.82 + 1.58i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.70 - 4.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-3 - 1.24i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.53 + 1.46i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.12 - 7.53i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.41 + 3.41i)T + 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (-4.24 - 4.24i)T + 59iT^{2} \)
61 \( 1 + (3.53 + 8.53i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 + (-12.0 - 5i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.05 + 4.94i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.82 - 1.58i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.242 + 0.242i)T - 83iT^{2} \)
89 \( 1 - 9.41iT - 89T^{2} \)
97 \( 1 + (2.46 - 5.94i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.31821713536500367241611526621, −9.507230792455593963361963910759, −8.722365039803538342355802105012, −8.117711069670278514875966122574, −7.33478196704110848616425145379, −6.64581974206498438238453114716, −4.93539282463832953866920602320, −3.76796090673214774285196824642, −3.24087127467736072981153171250, −2.34026749273806413543015666286, 0.866798517793666605561728992411, 2.04373010089125327509577518112, 3.06754158597410707986947932420, 4.37690558789050097573163237976, 5.42793661672039127517851221359, 6.51667080503974047620375036563, 7.61315191331281549305498076311, 8.380538882796718508737353717631, 8.903457004671655692780876433195, 9.686160137985528292510646442323

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