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Properties

Label 2-833-17.15-c1-0-38
Degree $2$
Conductor $833$
Sign $-0.323 - 0.946i$
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  + (1.33 + 1.33i)2-s + (0.902 + 2.17i)3-s + 1.57i·4-s + (3.13 − 1.29i)5-s + (−1.70 + 4.12i)6-s + (0.564 − 0.564i)8-s + (−1.81 + 1.81i)9-s + (5.92 + 2.45i)10-s + (−1.30 + 3.14i)11-s + (−3.43 + 1.42i)12-s − 3.48i·13-s + (5.65 + 5.65i)15-s + 4.66·16-s + (−3.82 − 1.53i)17-s − 4.85·18-s + (−0.765 − 0.765i)19-s + ⋯
L(s)  = 1  + (0.945 + 0.945i)2-s + (0.521 + 1.25i)3-s + 0.788i·4-s + (1.40 − 0.580i)5-s + (−0.697 + 1.68i)6-s + (0.199 − 0.199i)8-s + (−0.604 + 0.604i)9-s + (1.87 + 0.775i)10-s + (−0.393 + 0.949i)11-s + (−0.992 + 0.411i)12-s − 0.965i·13-s + (1.45 + 1.45i)15-s + 1.16·16-s + (−0.928 − 0.371i)17-s − 1.14·18-s + (−0.175 − 0.175i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.323 - 0.946i$
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: \((2,\ 833,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14660 + 3.00170i\)
\(L(\frac12)\) \(\approx\) \(2.14660 + 3.00170i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.82 + 1.53i)T \)
good2 \( 1 + (-1.33 - 1.33i)T + 2iT^{2} \)
3 \( 1 + (-0.902 - 2.17i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.13 + 1.29i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (1.30 - 3.14i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.48iT - 13T^{2} \)
19 \( 1 + (0.765 + 0.765i)T + 19iT^{2} \)
23 \( 1 + (1.87 - 4.52i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (6.71 - 2.78i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.30 + 5.56i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.50 - 3.64i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.21 + 1.74i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.546 + 0.546i)T - 43iT^{2} \)
47 \( 1 + 1.53iT - 47T^{2} \)
53 \( 1 + (6.00 + 6.00i)T + 53iT^{2} \)
59 \( 1 + (-7.75 + 7.75i)T - 59iT^{2} \)
61 \( 1 + (1.95 + 0.810i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 5.54T + 67T^{2} \)
71 \( 1 + (4.22 + 10.2i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (12.3 - 5.10i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.945 - 2.28i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.59 - 3.59i)T + 83iT^{2} \)
89 \( 1 - 5.49iT - 89T^{2} \)
97 \( 1 + (-9.52 + 3.94i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.02392882698663791734096859712, −9.756424467591015967541140912693, −8.934277998929454445463150298074, −7.83560842622079957192513943559, −6.80210190687916946179608995500, −5.70435435345567166882056829335, −5.16483127606202127318074312285, −4.49700846838620945953171049003, −3.43856665960712384932586990523, −2.01846146034690896456651305346, 1.65122157303874293487009418442, 2.23424100754435382447320302104, 3.02863009176725062760272693861, 4.32349884374601996429697079998, 5.66707863640195736628371899147, 6.32087275036312837331651527428, 7.18832681886780991244757792791, 8.303400567326794565374301598405, 9.114721138832059085193565050080, 10.30494075314876017248520595550

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