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Properties

Label 2-798-133.12-c1-0-0
Degree $2$
Conductor $798$
Sign $-0.981 + 0.190i$
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  + i·2-s + (0.5 − 0.866i)3-s − 4-s + 4.07i·5-s + (0.866 + 0.5i)6-s + (−1.41 + 2.23i)7-s − i·8-s + (−0.499 − 0.866i)9-s − 4.07·10-s + (−0.485 − 0.840i)11-s + (−0.5 + 0.866i)12-s + (−1.85 + 3.21i)13-s + (−2.23 − 1.41i)14-s + (3.52 + 2.03i)15-s + 16-s + (−6.48 − 3.74i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.288 − 0.499i)3-s − 0.5·4-s + 1.82i·5-s + (0.353 + 0.204i)6-s + (−0.533 + 0.845i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s − 1.28·10-s + (−0.146 − 0.253i)11-s + (−0.144 + 0.249i)12-s + (−0.514 + 0.891i)13-s + (−0.597 − 0.377i)14-s + (0.911 + 0.526i)15-s + 0.250·16-s + (−1.57 − 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.981 + 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0819147 - 0.853294i\)
\(L(\frac12)\) \(\approx\) \(0.0819147 - 0.853294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.41 - 2.23i)T \)
19 \( 1 + (-3.91 + 1.91i)T \)
good5 \( 1 - 4.07iT - 5T^{2} \)
11 \( 1 + (0.485 + 0.840i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.85 - 3.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.48 + 3.74i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.25 + 2.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.39 - 1.38i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.648 + 1.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.55 - 4.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.12 + 8.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.80 - 8.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.30 - 1.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + (4.75 - 8.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.9 - 6.88i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + 7.12iT - 67T^{2} \)
71 \( 1 + (0.430 - 0.248i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.85 - 1.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + (0.360 + 0.624i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.84 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.74305626393815806031968455926, −9.560949362625997591882532592566, −9.118077607125440442545362325328, −7.901841490068547109620012049182, −7.02126479064624669738854999526, −6.63135023533547091064122599774, −5.85429495506966292134181895458, −4.43911107600699354236657056428, −2.97265768401173370729402211836, −2.45484239416189271450168727682, 0.39485826140286264172380016406, 1.82489633098090233801597313732, 3.39535570125393798649022989907, 4.35231220912889907949774104653, 4.92447140980242017920340610741, 6.01409167241975504632219572650, 7.62682925432202176539270463362, 8.315846677067710404908082262837, 9.189853399205839115479300776022, 9.783649298100971474532970862392

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