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Properties

Label 2-833-17.15-c1-0-46
Degree $2$
Conductor $833$
Sign $0.00949 - 0.999i$
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  + (−1.65 − 1.65i)2-s + (−0.945 − 2.28i)3-s + 3.50i·4-s + (0.904 − 0.374i)5-s + (−2.21 + 5.35i)6-s + (2.50 − 2.50i)8-s + (−2.19 + 2.19i)9-s + (−2.12 − 0.878i)10-s + (1.35 − 3.26i)11-s + (8.00 − 3.31i)12-s − 6.30i·13-s + (−1.70 − 1.70i)15-s − 1.29·16-s + (−3.10 − 2.71i)17-s + 7.27·18-s + (1.07 + 1.07i)19-s + ⋯
L(s)  = 1  + (−1.17 − 1.17i)2-s + (−0.545 − 1.31i)3-s + 1.75i·4-s + (0.404 − 0.167i)5-s + (−0.905 + 2.18i)6-s + (0.885 − 0.885i)8-s + (−0.730 + 0.730i)9-s + (−0.670 − 0.277i)10-s + (0.407 − 0.984i)11-s + (2.31 − 0.957i)12-s − 1.74i·13-s + (−0.441 − 0.441i)15-s − 0.323·16-s + (−0.752 − 0.658i)17-s + 1.71·18-s + (0.247 + 0.247i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00949 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00949 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.00949 - 0.999i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.00949 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344749 + 0.341489i\)
\(L(\frac12)\) \(\approx\) \(0.344749 + 0.341489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.10 + 2.71i)T \)
good2 \( 1 + (1.65 + 1.65i)T + 2iT^{2} \)
3 \( 1 + (0.945 + 2.28i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.904 + 0.374i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (-1.35 + 3.26i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 6.30iT - 13T^{2} \)
19 \( 1 + (-1.07 - 1.07i)T + 19iT^{2} \)
23 \( 1 + (-0.124 + 0.301i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.28 + 0.533i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.387 - 0.935i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.48 + 3.57i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.84 - 2.00i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (8.26 - 8.26i)T - 43iT^{2} \)
47 \( 1 + 9.59iT - 47T^{2} \)
53 \( 1 + (-5.67 - 5.67i)T + 53iT^{2} \)
59 \( 1 + (-6.52 + 6.52i)T - 59iT^{2} \)
61 \( 1 + (-3.21 - 1.33i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.83T + 67T^{2} \)
71 \( 1 + (4.23 + 10.2i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.798 - 0.330i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.15 - 10.0i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.01 - 5.01i)T + 83iT^{2} \)
89 \( 1 - 0.265iT - 89T^{2} \)
97 \( 1 + (-6.43 + 2.66i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.690108477300040861842725635388, −8.743830686459834636151586455511, −8.050553959550162649754366380150, −7.30137655120071845666374096909, −6.17192149362104568508664318145, −5.38479200859225152897504823357, −3.45014158770572782733786583967, −2.43906559993184842568978251151, −1.27446917641027458756958243693, −0.41329991996831908489314064626, 1.86542325691163069335558599421, 4.02951607016726562187954400013, 4.75496755973058204752858391333, 5.86026102248320997759735274264, 6.61711496289973629247689422162, 7.28051476361798887240131410324, 8.634930609806901189632876987291, 9.170999098514432812045663103210, 9.893876692919794115279979303987, 10.29560422770341022599486480678

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