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Properties

Label 2-798-133.117-c1-0-16
Degree $2$
Conductor $798$
Sign $-0.218 - 0.975i$
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  + (0.342 + 0.939i)2-s + (0.766 + 0.642i)3-s + (−0.766 + 0.642i)4-s + (−0.636 + 0.758i)5-s + (−0.342 + 0.939i)6-s + (2.63 + 0.194i)7-s + (−0.866 − 0.500i)8-s + (0.173 + 0.984i)9-s + (−0.930 − 0.338i)10-s + 3.02·11-s − 12-s + (5.58 + 2.03i)13-s + (0.720 + 2.54i)14-s + (−0.974 + 0.171i)15-s + (0.173 − 0.984i)16-s + (−1.37 − 0.242i)17-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (0.442 + 0.371i)3-s + (−0.383 + 0.321i)4-s + (−0.284 + 0.339i)5-s + (−0.139 + 0.383i)6-s + (0.997 + 0.0733i)7-s + (−0.306 − 0.176i)8-s + (0.0578 + 0.328i)9-s + (−0.294 − 0.107i)10-s + 0.910·11-s − 0.288·12-s + (1.54 + 0.563i)13-s + (0.192 + 0.680i)14-s + (−0.251 + 0.0443i)15-s + (0.0434 − 0.246i)16-s + (−0.334 − 0.0589i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35473 + 1.69138i\)
\(L(\frac12)\) \(\approx\) \(1.35473 + 1.69138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-2.63 - 0.194i)T \)
19 \( 1 + (1.05 + 4.22i)T \)
good5 \( 1 + (0.636 - 0.758i)T + (-0.868 - 4.92i)T^{2} \)
11 \( 1 - 3.02T + 11T^{2} \)
13 \( 1 + (-5.58 - 2.03i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.37 + 0.242i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.59 + 2.03i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.50 - 1.79i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.79 - 4.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.19 + 1.26i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.80 - 2.47i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.21 - 6.90i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.70 + 1.18i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.69 + 5.59i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.265 - 1.50i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.75 + 10.3i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.18 + 3.26i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (7.98 + 1.40i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (-1.35 + 1.61i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-8.81 - 1.55i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.46 + 2.00i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-13.8 + 11.5i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-4.48 - 3.75i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.65830722313930067357023686270, −9.301688081525081897613059378784, −8.706371647253467773878463042677, −8.075922916104159060160917321836, −6.97888302588661091932295332921, −6.29053555565520387409297421534, −5.04387246054099718631284829166, −4.18507406722354896613227874046, −3.37995031971956117960106936097, −1.70537956302722035886885354873, 1.12713327803136312119177573092, 2.08828927082058788174139230169, 3.72252478522228747977492649152, 4.15494723533120672905528230932, 5.55449246514918974462896596736, 6.39453878812082818614898988824, 7.77210012093412249747044514496, 8.412694037517452793056873560947, 9.003042601691909811321421538764, 10.21355359924350308442764732263

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