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Properties

Label 2-833-17.15-c1-0-34
Degree $2$
Conductor $833$
Sign $-0.869 - 0.493i$
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  + (−1.61 − 1.61i)2-s + (−0.678 − 1.63i)3-s + 3.21i·4-s + (1.74 − 0.721i)5-s + (−1.55 + 3.74i)6-s + (1.96 − 1.96i)8-s + (−0.102 + 0.102i)9-s + (−3.97 − 1.64i)10-s + (0.942 − 2.27i)11-s + (5.27 − 2.18i)12-s + 3.98i·13-s + (−2.36 − 2.36i)15-s + 0.0862·16-s + (3.87 − 1.39i)17-s + 0.332·18-s + (−3.97 − 3.97i)19-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + (−0.391 − 0.946i)3-s + 1.60i·4-s + (0.778 − 0.322i)5-s + (−0.632 + 1.52i)6-s + (0.694 − 0.694i)8-s + (−0.0343 + 0.0343i)9-s + (−1.25 − 0.521i)10-s + (0.284 − 0.685i)11-s + (1.52 − 0.630i)12-s + 1.10i·13-s + (−0.610 − 0.610i)15-s + 0.0215·16-s + (0.940 − 0.338i)17-s + 0.0783·18-s + (−0.910 − 0.910i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.869 - 0.493i$
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: \((2,\ 833,\ (\ :1/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182269 + 0.690644i\)
\(L(\frac12)\) \(\approx\) \(0.182269 + 0.690644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.87 + 1.39i)T \)
good2 \( 1 + (1.61 + 1.61i)T + 2iT^{2} \)
3 \( 1 + (0.678 + 1.63i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.74 + 0.721i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (-0.942 + 2.27i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.98iT - 13T^{2} \)
19 \( 1 + (3.97 + 3.97i)T + 19iT^{2} \)
23 \( 1 + (-2.23 + 5.39i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.946 - 0.391i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.859 + 2.07i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.0997 + 0.240i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.97 + 2.88i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-6.88 + 6.88i)T - 43iT^{2} \)
47 \( 1 + 8.05iT - 47T^{2} \)
53 \( 1 + (-1.54 - 1.54i)T + 53iT^{2} \)
59 \( 1 + (5.29 - 5.29i)T - 59iT^{2} \)
61 \( 1 + (13.0 + 5.41i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (-4.40 - 10.6i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.47 - 1.44i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.15 - 2.79i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-8.14 - 8.14i)T + 83iT^{2} \)
89 \( 1 - 8.47iT - 89T^{2} \)
97 \( 1 + (-2.92 + 1.21i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.632785920014976121957667714918, −9.071375289514700825340188550784, −8.365371634234497891757750729922, −7.22075556380482923861447773974, −6.46317333650478535192408692753, −5.43450365282254862650119187388, −3.89623323455561581667687733142, −2.46766514623386247517102828557, −1.61427861334112371517587781673, −0.56882208449414399726559435231, 1.58037797041053318523305203778, 3.45729054517359160187038113359, 4.85199002084650408080681621844, 5.76645172635278280443405379018, 6.27129938311233931919857244079, 7.50402923375670155416442496844, 8.019794425971083538196465736429, 9.232880092646184585050795341908, 9.789923246439708658692909507486, 10.31418751573136425119799207504

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