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Properties

Label 2-798-133.100-c1-0-11
Degree $2$
Conductor $798$
Sign $0.982 - 0.188i$
Analytic cond. $6.37206$
Root an. cond. $2.52429$
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  + (−0.173 + 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−4.09 + 1.49i)5-s + (0.173 + 0.984i)6-s + (−1.82 − 1.91i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.756 − 4.29i)10-s + 4.08·11-s − 0.999·12-s + (−0.160 − 0.912i)13-s + (2.20 − 1.46i)14-s + (−3.33 + 2.80i)15-s + (0.766 + 0.642i)16-s + (3.22 + 2.70i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.542 − 0.197i)3-s + (−0.469 − 0.171i)4-s + (−1.83 + 0.666i)5-s + (0.0708 + 0.402i)6-s + (−0.689 − 0.724i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.239 − 1.35i)10-s + 1.23·11-s − 0.288·12-s + (−0.0446 − 0.253i)13-s + (0.589 − 0.391i)14-s + (−0.862 + 0.723i)15-s + (0.191 + 0.160i)16-s + (0.783 + 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11918 + 0.106577i\)
\(L(\frac12)\) \(\approx\) \(1.11918 + 0.106577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (1.82 + 1.91i)T \)
19 \( 1 + (-2.71 + 3.40i)T \)
good5 \( 1 + (4.09 - 1.49i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 - 4.08T + 11T^{2} \)
13 \( 1 + (0.160 + 0.912i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.22 - 2.70i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.473 + 2.68i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-7.00 - 2.54i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.72 - 6.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.960 + 1.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0781 + 0.443i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.42 + 1.19i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.01 + 5.04i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-13.0 - 4.75i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (9.81 + 8.23i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.83 + 10.4i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.77 + 10.0i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (5.73 + 4.81i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-14.6 + 5.34i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.28 + 1.07i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.83 + 2.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.11 + 1.49i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.31994749721855651459406587112, −9.176337198843964577708943560359, −8.403732901279326790471823528682, −7.58906318147879127898078809492, −6.98996916899874498295586868320, −6.42593371228398037840813518390, −4.67919862089282967166783580096, −3.71331841695079773656281101275, −3.23588106983192691921486383533, −0.75540294064075537125476190580, 1.05362693383572794256994794745, 2.88765561809169089329305451500, 3.76248266739684380306911148530, 4.33675522749974309875415083164, 5.62148102722523938941500860336, 7.12350166265252870507714388044, 7.898668122446866141090223263777, 8.727653633310318209399344044747, 9.290803395650704426705345083805, 10.06451406899206065571728849333

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