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Properties

Label 2-833-17.8-c1-0-33
Degree $2$
Conductor $833$
Sign $0.522 + 0.852i$
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  + (−0.410 + 0.410i)2-s + (0.115 − 0.277i)3-s + 1.66i·4-s + (−2.34 − 0.970i)5-s + (0.0667 + 0.161i)6-s + (−1.50 − 1.50i)8-s + (2.05 + 2.05i)9-s + (1.36 − 0.563i)10-s + (−0.487 − 1.17i)11-s + (0.461 + 0.191i)12-s − 4.93i·13-s + (−0.539 + 0.539i)15-s − 2.09·16-s + (1.26 − 3.92i)17-s − 1.68·18-s + (−5.34 + 5.34i)19-s + ⋯
L(s)  = 1  + (−0.290 + 0.290i)2-s + (0.0664 − 0.160i)3-s + 0.831i·4-s + (−1.04 − 0.434i)5-s + (0.0272 + 0.0658i)6-s + (−0.531 − 0.531i)8-s + (0.685 + 0.685i)9-s + (0.430 − 0.178i)10-s + (−0.146 − 0.354i)11-s + (0.133 + 0.0552i)12-s − 1.36i·13-s + (−0.139 + 0.139i)15-s − 0.522·16-s + (0.306 − 0.951i)17-s − 0.398·18-s + (−1.22 + 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689720 - 0.386051i\)
\(L(\frac12)\) \(\approx\) \(0.689720 - 0.386051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 + 3.92i)T \)
good2 \( 1 + (0.410 - 0.410i)T - 2iT^{2} \)
3 \( 1 + (-0.115 + 0.277i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (2.34 + 0.970i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.487 + 1.17i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.93iT - 13T^{2} \)
19 \( 1 + (5.34 - 5.34i)T - 19iT^{2} \)
23 \( 1 + (0.597 + 1.44i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-6.01 - 2.49i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.67 + 4.04i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.05 + 7.38i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-10.2 + 4.23i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.58 + 4.58i)T + 43iT^{2} \)
47 \( 1 + 7.57iT - 47T^{2} \)
53 \( 1 + (-1.54 + 1.54i)T - 53iT^{2} \)
59 \( 1 + (-2.28 - 2.28i)T + 59iT^{2} \)
61 \( 1 + (2.24 - 0.928i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 9.97T + 67T^{2} \)
71 \( 1 + (-3.12 + 7.53i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.94 + 1.63i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.16 - 10.0i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-4.67 + 4.67i)T - 83iT^{2} \)
89 \( 1 + 2.46iT - 89T^{2} \)
97 \( 1 + (13.2 + 5.50i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.12636891716071816012004420233, −8.895453055489731829908107555725, −8.026655619795006841216376841555, −7.87797888900991735922797540916, −6.96508659321783189716231867304, −5.71629677363778883336481301570, −4.48706951232157196809076687198, −3.72784697637891425403980479170, −2.54792383199652476526960213008, −0.46267857294533503088749911825, 1.33015072841947069303178794787, 2.74014133947937996513529586679, 4.16157062219271759865287540297, 4.64501333892055879205392126536, 6.35141454875526127352727436890, 6.69522266661511538836041738020, 7.87737392126543182627649671947, 8.849024751066444818610374363905, 9.578631166934329589356548467702, 10.35403138627924483781238692238

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