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Properties

Label 2-798-399.107-c1-0-42
Degree $2$
Conductor $798$
Sign $0.869 + 0.494i$
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 + (1.38 + 1.03i)3-s + 4-s − 2.72i·5-s + (1.38 + 1.03i)6-s + (−1.73 − 2.00i)7-s + 8-s + (0.858 + 2.87i)9-s − 2.72i·10-s + (4.60 − 2.65i)11-s + (1.38 + 1.03i)12-s + (−3.44 − 1.99i)13-s + (−1.73 − 2.00i)14-s + (2.82 − 3.78i)15-s + 16-s + (2.99 + 1.72i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.801 + 0.597i)3-s + 0.5·4-s − 1.21i·5-s + (0.567 + 0.422i)6-s + (−0.654 − 0.756i)7-s + 0.353·8-s + (0.286 + 0.958i)9-s − 0.862i·10-s + (1.38 − 0.800i)11-s + (0.400 + 0.298i)12-s + (−0.956 − 0.552i)13-s + (−0.462 − 0.534i)14-s + (0.728 − 0.977i)15-s + 0.250·16-s + (0.726 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.869 + 0.494i$
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: \((2,\ 798,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89950 - 0.766594i\)
\(L(\frac12)\) \(\approx\) \(2.89950 - 0.766594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.03i)T \)
7 \( 1 + (1.73 + 2.00i)T \)
19 \( 1 + (2.29 - 3.70i)T \)
good5 \( 1 + 2.72iT - 5T^{2} \)
11 \( 1 + (-4.60 + 2.65i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.44 + 1.99i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.99 - 1.72i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.179 - 0.103i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.95 + 6.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.34 + 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.46 + 1.42i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.64 - 4.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.50 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.98 - 5.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.20T + 53T^{2} \)
59 \( 1 + (-1.18 + 2.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.84 - 8.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 5.84iT - 67T^{2} \)
71 \( 1 + (-3.95 - 6.85i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.47 - 6.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.0iT - 79T^{2} \)
83 \( 1 - 3.60iT - 83T^{2} \)
89 \( 1 + (4.66 + 8.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 - 6.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.908087705215258324445168299711, −9.584321231952631580736025461608, −8.387727813105037371956412831619, −7.87702714252358012806779062688, −6.57122291890727536045976715804, −5.61032995356548149101460083934, −4.45958970416296763148830864910, −3.94586434086164342114476522628, −2.92794162871994203378848725967, −1.24095417412154572032063486269, 1.94540013886332297273502140567, 2.84014036604990032852718609949, 3.58968640234847375714376025410, 4.86119459254822668926043137859, 6.36989555469533673558800306576, 6.81819087885097202782622152203, 7.31376636337014871573078605694, 8.719728427700051879068031322527, 9.489093378811354656670922444323, 10.22167711167718704470874461457

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