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Properties

Label 2-798-133.10-c1-0-11
Degree $2$
Conductor $798$
Sign $0.826 - 0.563i$
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  + (−0.642 − 0.766i)2-s + (0.173 + 0.984i)3-s + (−0.173 + 0.984i)4-s + (2.83 − 0.499i)5-s + (0.642 − 0.766i)6-s + (−1.47 + 2.19i)7-s + (0.866 − 0.500i)8-s + (−0.939 + 0.342i)9-s + (−2.20 − 1.85i)10-s + 2.19·11-s − 12-s + (0.145 + 0.122i)13-s + (2.63 − 0.279i)14-s + (0.984 + 2.70i)15-s + (−0.939 − 0.342i)16-s + (0.626 − 1.71i)17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.100 + 0.568i)3-s + (−0.0868 + 0.492i)4-s + (1.26 − 0.223i)5-s + (0.262 − 0.312i)6-s + (−0.558 + 0.829i)7-s + (0.306 − 0.176i)8-s + (−0.313 + 0.114i)9-s + (−0.697 − 0.585i)10-s + 0.660·11-s − 0.288·12-s + (0.0404 + 0.0339i)13-s + (0.703 − 0.0747i)14-s + (0.254 + 0.698i)15-s + (−0.234 − 0.0855i)16-s + (0.151 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.826 - 0.563i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43059 + 0.441267i\)
\(L(\frac12)\) \(\approx\) \(1.43059 + 0.441267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (1.47 - 2.19i)T \)
19 \( 1 + (-2.78 - 3.35i)T \)
good5 \( 1 + (-2.83 + 0.499i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + (-0.145 - 0.122i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.626 + 1.71i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.58 - 3.01i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.861 + 0.151i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.778 + 1.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 - 2.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.37 - 5.34i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.85 - 1.40i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.74 - 7.55i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.69 - 1.00i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-0.954 - 0.347i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.50 + 2.98i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-9.18 + 10.9i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.04 - 2.86i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.56 + 0.629i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.61 - 7.19i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.93 - 4.00i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.25 - 7.14i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.933 - 5.29i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.01529255489597492604335098310, −9.505453522140553831841081010167, −9.133080066899807078442872473132, −8.132688782992948502283417974474, −6.79996981141341214853347860304, −5.81837065865945614810581815088, −5.09430450440605100172123611544, −3.65083833945122330498490897598, −2.68273881138890373141892362160, −1.49993033388144418299497741741, 0.954083882582958948815018603980, 2.23444955760042770359868830970, 3.63243935796127451729513017995, 5.15311117019623719939177735046, 6.06891719793492100674713792022, 6.85143478796883697017128033917, 7.28200650974930895073915394724, 8.640687525950746493217313308651, 9.239499694559999910197440060363, 10.12664449137562116097663794056

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