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Properties

Label 2-798-133.123-c1-0-9
Degree $2$
Conductor $798$
Sign $0.664 - 0.746i$
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  + (−0.766 + 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.132 − 0.752i)5-s + (−0.766 − 0.642i)6-s + (2.46 − 0.961i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.585 + 0.491i)10-s − 4.18·11-s + 12-s + (0.739 + 0.620i)13-s + (−1.27 + 2.32i)14-s + (0.718 − 0.261i)15-s + (−0.939 − 0.342i)16-s + (2.33 + 0.850i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.100 + 0.568i)3-s + (0.0868 − 0.492i)4-s + (−0.0593 − 0.336i)5-s + (−0.312 − 0.262i)6-s + (0.931 − 0.363i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.185 + 0.155i)10-s − 1.26·11-s + 0.288·12-s + (0.205 + 0.172i)13-s + (−0.339 + 0.620i)14-s + (0.185 − 0.0675i)15-s + (−0.234 − 0.0855i)16-s + (0.566 + 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.664 - 0.746i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.664 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18430 + 0.531343i\)
\(L(\frac12)\) \(\approx\) \(1.18430 + 0.531343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-2.46 + 0.961i)T \)
19 \( 1 + (-4.08 - 1.50i)T \)
good5 \( 1 + (0.132 + 0.752i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
13 \( 1 + (-0.739 - 0.620i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.33 - 0.850i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.51 - 4.62i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.887 + 5.03i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (4.14 + 7.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.51 - 9.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.27 - 1.07i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-9.28 - 3.37i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.36 + 0.496i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-0.349 + 1.98i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-7.70 - 2.80i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.87 - 2.41i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.953 + 0.799i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (9.44 + 3.43i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.38 - 7.83i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (11.2 + 4.11i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (8.72 - 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.36 - 13.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.82 + 16.0i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.19041602737143642269655869825, −9.590699982439575261985449109346, −8.560238508757165309110351917924, −7.892617466246055102479522524734, −7.28975093795256121545290101468, −5.78012722129829444671920586365, −5.15007384675911244885305950709, −4.21756740944955913877504912356, −2.76098833073177177319722409570, −1.12307016916000215562631267810, 1.02145376177704951094896187319, 2.42719059097729091825153166840, 3.19365190214191629965211092581, 4.87376383390712923711117833787, 5.64545087115483863584607041312, 7.20341789509850659467575639140, 7.48397278968026131399343364056, 8.587872655791968388937357905973, 9.067735508100158423565215610157, 10.46253424935533078233032241951

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