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Properties

Label 2-798-133.16-c1-0-14
Degree $2$
Conductor $798$
Sign $0.862 - 0.506i$
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  + (0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (1.46 + 1.22i)5-s + (0.939 + 0.342i)6-s + (1.44 + 2.21i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (1.79 + 0.653i)10-s + 0.411·11-s + 12-s + (−1.80 − 0.658i)13-s + (2.11 + 1.58i)14-s + (0.331 + 1.88i)15-s + (0.173 − 0.984i)16-s + (0.494 − 2.80i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.442 + 0.371i)3-s + (0.383 − 0.321i)4-s + (0.654 + 0.548i)5-s + (0.383 + 0.139i)6-s + (0.546 + 0.837i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.567 + 0.206i)10-s + 0.124·11-s + 0.288·12-s + (−0.501 − 0.182i)13-s + (0.565 + 0.424i)14-s + (0.0856 + 0.485i)15-s + (0.0434 − 0.246i)16-s + (0.119 − 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.862 - 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91243 + 0.791470i\)
\(L(\frac12)\) \(\approx\) \(2.91243 + 0.791470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-1.44 - 2.21i)T \)
19 \( 1 + (2.63 - 3.46i)T \)
good5 \( 1 + (-1.46 - 1.22i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 - 0.411T + 11T^{2} \)
13 \( 1 + (1.80 + 0.658i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.494 + 2.80i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.34 - 0.852i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (4.28 - 3.59i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.756 + 1.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.86 + 3.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.92 - 1.06i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.61 + 9.17i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.10 - 6.24i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-10.8 + 9.06i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-2.04 + 11.5i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.24 + 0.451i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.520 - 0.189i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (1.56 - 8.88i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.294 + 0.247i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.08 + 17.4i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.36 - 4.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.15 - 6.84i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (3.12 + 2.62i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.36654677562690162754233998445, −9.617693376180014669047360700800, −8.790041526047131509716238014094, −7.75288975711142276385482301042, −6.73154411505825192422749887985, −5.68370588451508843772741063867, −5.04733212488441142336315433922, −3.84523935583038239118325823514, −2.70502788742751393264220746619, −1.95391084010634353520904634383, 1.35944489483253863773234720508, 2.52705057742927067155345379786, 3.93707779468541789362816345823, 4.74503308694154417912637537301, 5.73870715000288532811449695615, 6.76484116691087738473723700643, 7.49540258818553668574492913247, 8.402302677107650382731481242522, 9.227356344412652318053148648302, 10.21671616187119317049237760887

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