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Properties

Label 2-798-133.10-c1-0-10
Degree $2$
Conductor $798$
Sign $-0.329 - 0.944i$
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  + (0.642 + 0.766i)2-s + (0.173 + 0.984i)3-s + (−0.173 + 0.984i)4-s + (1.71 − 0.302i)5-s + (−0.642 + 0.766i)6-s + (−2.45 − 0.976i)7-s + (−0.866 + 0.500i)8-s + (−0.939 + 0.342i)9-s + (1.33 + 1.12i)10-s + 4.17·11-s − 12-s + (1.65 + 1.38i)13-s + (−0.832 − 2.51i)14-s + (0.596 + 1.63i)15-s + (−0.939 − 0.342i)16-s + (−2.61 + 7.19i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.100 + 0.568i)3-s + (−0.0868 + 0.492i)4-s + (0.768 − 0.135i)5-s + (−0.262 + 0.312i)6-s + (−0.929 − 0.369i)7-s + (−0.306 + 0.176i)8-s + (−0.313 + 0.114i)9-s + (0.422 + 0.354i)10-s + 1.26·11-s − 0.288·12-s + (0.458 + 0.384i)13-s + (−0.222 − 0.671i)14-s + (0.154 + 0.423i)15-s + (−0.234 − 0.0855i)16-s + (−0.635 + 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21032 + 1.70423i\)
\(L(\frac12)\) \(\approx\) \(1.21032 + 1.70423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (2.45 + 0.976i)T \)
19 \( 1 + (-0.929 - 4.25i)T \)
good5 \( 1 + (-1.71 + 0.302i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
13 \( 1 + (-1.65 - 1.38i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.61 - 7.19i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.56 - 3.83i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.14 - 0.202i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.23 + 2.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.370 - 0.213i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.91 + 4.96i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.89 + 1.05i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (3.51 + 9.67i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (12.3 + 2.18i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-8.47 - 3.08i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.55 + 6.61i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.28 + 2.72i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.09 + 11.2i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-12.0 + 2.11i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-3.65 + 10.0i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (13.2 + 7.65i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.72 - 15.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.13 - 17.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.39471448552968463115581599549, −9.493459739193587724632741701323, −9.029593610389454457786790490273, −7.992007128572045594685075517887, −6.65263130681530593597244207550, −6.25197986836476129058115639016, −5.32594954378200039116751696583, −3.91481252987880587873192206442, −3.61459146376346708320402125382, −1.78042666832084024176197054558, 0.949345716292344083787365954515, 2.46857340906245380260667809643, 3.13445489799010956500535132326, 4.54955362613892790412981794801, 5.65730161192558065424249946248, 6.56521813355730269671099491694, 6.97153827056022381740048939155, 8.611657324600975905950273348815, 9.388773906920529433379640389791, 9.792563692983964269434803524330

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