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Properties

Label 2-798-133.75-c1-0-3
Degree $2$
Conductor $798$
Sign $0.756 - 0.654i$
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  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−3.04 − 1.75i)5-s + 0.999i·6-s + (−0.775 + 2.52i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.75 + 3.04i)10-s + (−3.11 − 5.39i)11-s + (0.499 − 0.866i)12-s + 1.71·13-s + (1.93 − 1.80i)14-s + 3.51i·15-s + (−0.5 + 0.866i)16-s + (−2.10 + 1.21i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−1.35 − 0.785i)5-s + 0.408i·6-s + (−0.293 + 0.956i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.555 + 0.961i)10-s + (−0.939 − 1.62i)11-s + (0.144 − 0.249i)12-s + 0.474·13-s + (0.517 − 0.481i)14-s + 0.906i·15-s + (−0.125 + 0.216i)16-s + (−0.510 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.756 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318599 + 0.118735i\)
\(L(\frac12)\) \(\approx\) \(0.318599 + 0.118735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.775 - 2.52i)T \)
19 \( 1 + (-1.82 + 3.95i)T \)
good5 \( 1 + (3.04 + 1.75i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.11 + 5.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
17 \( 1 + (2.10 - 1.21i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.08 - 5.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.45iT - 29T^{2} \)
31 \( 1 + (-3.26 - 5.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.63 + 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-9.47 - 5.47i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.06 + 0.614i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.47 - 0.848i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.02 + 2.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.52iT - 71T^{2} \)
73 \( 1 + (8.30 - 4.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.76 + 2.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (-0.178 + 0.309i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−10.72638657071825125520176774320, −9.109010723976665725640969524182, −8.686264349137878699769297489929, −8.028432419150009383472449076160, −7.19656416526515749218185835699, −5.94419266840288836777583724581, −5.12668482440863826459385583440, −3.68943706013620798110895017865, −2.75823047477159159584304456929, −0.993626919397186388057239267901, 0.27320290299804785838998736610, 2.53668355092779621675028526806, 4.00465082185306802786233344093, 4.46927047685210644367528283137, 5.99861526496777400394574304013, 6.99402655834329888439664424737, 7.59214471734249871411843099088, 8.175713902784117934381902961833, 9.555545310017557094607134989246, 10.31929800322480875759236183687

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