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Properties

Label 2-798-399.107-c1-0-13
Degree $2$
Conductor $798$
Sign $0.697 - 0.716i$
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 + (−1.31 + 1.12i)3-s + 4-s − 2.51i·5-s + (−1.31 + 1.12i)6-s + (−0.461 + 2.60i)7-s + 8-s + (0.453 − 2.96i)9-s − 2.51i·10-s + (−2.23 + 1.28i)11-s + (−1.31 + 1.12i)12-s + (2.19 + 1.26i)13-s + (−0.461 + 2.60i)14-s + (2.84 + 3.30i)15-s + 16-s + (6.13 + 3.54i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.758 + 0.651i)3-s + 0.5·4-s − 1.12i·5-s + (−0.536 + 0.460i)6-s + (−0.174 + 0.984i)7-s + 0.353·8-s + (0.151 − 0.988i)9-s − 0.795i·10-s + (−0.672 + 0.388i)11-s + (−0.379 + 0.325i)12-s + (0.608 + 0.351i)13-s + (−0.123 + 0.696i)14-s + (0.733 + 0.853i)15-s + 0.250·16-s + (1.48 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.697 - 0.716i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71239 + 0.722225i\)
\(L(\frac12)\) \(\approx\) \(1.71239 + 0.722225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 - 1.12i)T \)
7 \( 1 + (0.461 - 2.60i)T \)
19 \( 1 + (-1.28 - 4.16i)T \)
good5 \( 1 + 2.51iT - 5T^{2} \)
11 \( 1 + (2.23 - 1.28i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.19 - 1.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.13 - 3.54i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.35 + 3.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.98 + 5.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.78 - 4.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.50 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.28 + 2.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.11 - 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.73 - 2.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + (-0.904 + 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.75 - 9.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 0.726iT - 67T^{2} \)
71 \( 1 + (2.85 + 4.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.99 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + (1.43 + 2.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.70 - 5.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.42094480787390166567554040296, −9.645138321420018249999156085497, −8.756387872764308504625978872975, −7.896244042957050933880302019916, −6.45930223225833749335347481759, −5.63828966265609026716842848090, −5.15525097186382551623491163159, −4.22320379396471118735490793810, −3.12268528117161763059903737873, −1.37106810395022880682663128982, 0.953519804909916946622418172530, 2.76046111105016827262162002493, 3.52263196361644526067349209572, 5.02406273618037675872879602734, 5.70505611782101621718144518416, 6.79415834469161944098680699837, 7.24904551415221512965891938189, 7.932257790750372523829463070144, 9.634518749879454054781635036422, 10.63414777231002277926764472868

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