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Properties

Label 2-798-399.179-c1-0-38
Degree $2$
Conductor $798$
Sign $0.0684 + 0.997i$
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  + 2-s + (−1.54 − 0.784i)3-s + 4-s − 3.23i·5-s + (−1.54 − 0.784i)6-s + (2.52 − 0.776i)7-s + 8-s + (1.76 + 2.42i)9-s − 3.23i·10-s + (3.60 + 2.07i)11-s + (−1.54 − 0.784i)12-s + (−2.79 + 1.61i)13-s + (2.52 − 0.776i)14-s + (−2.54 + 4.99i)15-s + 16-s + (5.17 − 2.98i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.891 − 0.453i)3-s + 0.5·4-s − 1.44i·5-s + (−0.630 − 0.320i)6-s + (0.955 − 0.293i)7-s + 0.353·8-s + (0.589 + 0.807i)9-s − 1.02i·10-s + (1.08 + 0.626i)11-s + (−0.445 − 0.226i)12-s + (−0.775 + 0.447i)13-s + (0.675 − 0.207i)14-s + (−0.656 + 1.29i)15-s + 0.250·16-s + (1.25 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0684 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0684 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.0684 + 0.997i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.0684 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48477 - 1.38644i\)
\(L(\frac12)\) \(\approx\) \(1.48477 - 1.38644i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.54 + 0.784i)T \)
7 \( 1 + (-2.52 + 0.776i)T \)
19 \( 1 + (0.893 + 4.26i)T \)
good5 \( 1 + 3.23iT - 5T^{2} \)
11 \( 1 + (-3.60 - 2.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.79 - 1.61i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.17 + 2.98i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.38 + 0.800i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.61 + 6.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0503 - 0.0290i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.49 - 4.90i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.30 - 9.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.151 - 0.262i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.40 - 0.812i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + (4.95 + 8.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.878 - 1.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 6.48iT - 67T^{2} \)
71 \( 1 + (-7.51 + 13.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.01 - 1.76i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.11iT - 79T^{2} \)
83 \( 1 - 6.21iT - 83T^{2} \)
89 \( 1 + (3.98 - 6.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 - 7.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12497419274842801111319609254, −9.303197406707465395567475356603, −8.135963850726519945845481040098, −7.34905727525107614395751250679, −6.50272885963996891420830563701, −5.21952074049537274227189863869, −4.89443084181088692393689823653, −4.10340850516305936240729630787, −1.98670755479152955878298989336, −1.00681913531655895225445759128, 1.73014909122294426857754393587, 3.35771070192908765241958898408, 3.95684602295392425515457993743, 5.42074535049325530791925185350, 5.78965334833050532100324746762, 6.86646871802544411593240792748, 7.52694303373487739020294169568, 8.800818617311857511622222024001, 10.20023775030415673340363097649, 10.51942421066089985120556568884

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