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Properties

Label 2-798-133.75-c1-0-9
Degree $2$
Conductor $798$
Sign $0.134 - 0.990i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−2.81 − 1.62i)5-s + 0.999i·6-s + (2.61 + 0.400i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.62 − 2.81i)10-s + (2.06 + 3.57i)11-s + (−0.499 + 0.866i)12-s + 2.67·13-s + (2.06 + 1.65i)14-s − 3.24i·15-s + (−0.5 + 0.866i)16-s + (−2.12 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−1.25 − 0.725i)5-s + 0.408i·6-s + (0.988 + 0.151i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.513 − 0.888i)10-s + (0.623 + 1.07i)11-s + (−0.144 + 0.249i)12-s + 0.741·13-s + (0.551 + 0.442i)14-s − 0.838i·15-s + (−0.125 + 0.216i)16-s + (−0.514 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.134 - 0.990i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.134 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64775 + 1.43898i\)
\(L(\frac12)\) \(\approx\) \(1.64775 + 1.43898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.61 - 0.400i)T \)
19 \( 1 + (-4.31 + 0.624i)T \)
good5 \( 1 + (2.81 + 1.62i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.67T + 13T^{2} \)
17 \( 1 + (2.12 - 1.22i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.24 - 3.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.43iT - 29T^{2} \)
31 \( 1 + (0.831 + 1.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.75 + 5.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.53T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (-9.65 - 5.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.72 - 5.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.12 + 5.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.5 + 6.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.27 + 4.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.08iT - 71T^{2} \)
73 \( 1 + (3.98 - 2.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.73 + 2.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.0iT - 83T^{2} \)
89 \( 1 + (-5.38 + 9.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.90T + 97T^{2} \)
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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

−10.80451129496345216584072149855, −9.215352458021347371626954027128, −8.803637146941729879125757339802, −7.68070930337160665552744577237, −7.37604716838212642600460515808, −5.82565943936551034454942633872, −4.79333177620645728936939682573, −4.26789476827165731240297353765, −3.43136955584410747166086460322, −1.65251891449630195000555058346, 0.983757137229165148382455354686, 2.57178724904664317062769708042, 3.65319199295356545217770617422, 4.26828085377289728563559882524, 5.67630316774736198964458279227, 6.62496090790224761128788658005, 7.52387526332008297087039798754, 8.209940487444974611478305801150, 9.039047267428303714613227777641, 10.52396810037661577082361875892

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