Properties

Label 2-799-47.21-c1-0-34
Degree $2$
Conductor $799$
Sign $-0.407 + 0.913i$
Analytic cond. $6.38004$
Root an. cond. $2.52587$
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  + (−0.573 − 1.10i)2-s + (−0.697 + 1.96i)3-s + (0.255 − 0.362i)4-s + (−1.00 − 0.436i)5-s + (2.57 − 0.353i)6-s + (−0.732 − 0.445i)7-s + (−3.02 − 0.415i)8-s + (−1.03 − 0.844i)9-s + (0.0933 + 1.36i)10-s + (4.39 − 1.23i)11-s + (0.532 + 0.754i)12-s + (0.0240 + 0.115i)13-s + (−0.0730 + 1.06i)14-s + (1.55 − 1.66i)15-s + (0.976 + 2.74i)16-s + (−0.962 − 0.269i)17-s + ⋯
L(s)  = 1  + (−0.405 − 0.783i)2-s + (−0.402 + 1.13i)3-s + (0.127 − 0.181i)4-s + (−0.449 − 0.195i)5-s + (1.05 − 0.144i)6-s + (−0.277 − 0.168i)7-s + (−1.06 − 0.146i)8-s + (−0.346 − 0.281i)9-s + (0.0295 + 0.431i)10-s + (1.32 − 0.370i)11-s + (0.153 + 0.217i)12-s + (0.00666 + 0.0320i)13-s + (−0.0195 + 0.285i)14-s + (0.402 − 0.430i)15-s + (0.244 + 0.686i)16-s + (−0.233 − 0.0654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.407 + 0.913i$
Analytic conductor: \(6.38004\)
Root analytic conductor: \(2.52587\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.414311 - 0.638292i\)
\(L(\frac12)\) \(\approx\) \(0.414311 - 0.638292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.962 + 0.269i)T \)
47 \( 1 + (-6.79 - 0.928i)T \)
good2 \( 1 + (0.573 + 1.10i)T + (-1.15 + 1.63i)T^{2} \)
3 \( 1 + (0.697 - 1.96i)T + (-2.32 - 1.89i)T^{2} \)
5 \( 1 + (1.00 + 0.436i)T + (3.41 + 3.65i)T^{2} \)
7 \( 1 + (0.732 + 0.445i)T + (3.22 + 6.21i)T^{2} \)
11 \( 1 + (-4.39 + 1.23i)T + (9.39 - 5.71i)T^{2} \)
13 \( 1 + (-0.0240 - 0.115i)T + (-11.9 + 5.17i)T^{2} \)
19 \( 1 + (0.551 - 0.239i)T + (12.9 - 13.8i)T^{2} \)
23 \( 1 + (1.03 - 2.00i)T + (-13.2 - 18.7i)T^{2} \)
29 \( 1 + (-0.874 + 4.20i)T + (-26.5 - 11.5i)T^{2} \)
31 \( 1 + (2.99 + 8.41i)T + (-24.0 + 19.5i)T^{2} \)
37 \( 1 + (0.683 + 9.99i)T + (-36.6 + 5.03i)T^{2} \)
41 \( 1 + (-5.03 + 0.691i)T + (39.4 - 11.0i)T^{2} \)
43 \( 1 + (-2.78 + 3.94i)T + (-14.3 - 40.5i)T^{2} \)
53 \( 1 + (-11.8 + 1.62i)T + (51.0 - 14.2i)T^{2} \)
59 \( 1 + (5.64 + 8.00i)T + (-19.7 + 55.5i)T^{2} \)
61 \( 1 + (-0.881 + 12.8i)T + (-60.4 - 8.30i)T^{2} \)
67 \( 1 + (4.57 - 2.77i)T + (30.8 - 59.4i)T^{2} \)
71 \( 1 + (2.50 - 4.83i)T + (-40.9 - 58.0i)T^{2} \)
73 \( 1 + (7.19 - 5.85i)T + (14.8 - 71.4i)T^{2} \)
79 \( 1 + (-0.557 + 0.596i)T + (-5.39 - 78.8i)T^{2} \)
83 \( 1 + (0.242 - 0.0680i)T + (70.9 - 43.1i)T^{2} \)
89 \( 1 + (3.52 + 1.53i)T + (60.7 + 65.0i)T^{2} \)
97 \( 1 + (-3.39 + 9.54i)T + (-75.2 - 61.2i)T^{2} \)
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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

−10.01786151219015243368041084192, −9.416911167455686593078099040358, −8.814678034223615502330283909246, −7.48992392859366688926388908698, −6.27139797193263836862021587990, −5.56772962167156258466963412167, −4.09565536985194243006811108537, −3.77515863539338553998433343678, −2.14865221763405630701145659539, −0.47937445779038820695418666461, 1.39890485682562427496401643089, 2.94131913355109463304375689171, 4.17490378149129097342659920367, 5.77350681039291213350647411912, 6.52021812921894801925903635548, 7.06113782599931436780640085116, 7.64928001201367791882070431662, 8.696800975759328295646870752055, 9.295002045028297606450412649890, 10.60430429392516590794201827413

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