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Properties

Label 2-798-133.103-c1-0-23
Degree $2$
Conductor $798$
Sign $0.482 + 0.875i$
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 + 3-s + (0.499 − 0.866i)4-s + (0.479 − 0.276i)5-s + (0.866 − 0.5i)6-s + (2.21 − 1.44i)7-s − 0.999i·8-s + 9-s + (0.276 − 0.479i)10-s + (−0.816 − 1.41i)11-s + (0.499 − 0.866i)12-s + (−0.372 − 0.645i)13-s + (1.19 − 2.35i)14-s + (0.479 − 0.276i)15-s + (−0.5 − 0.866i)16-s − 1.55i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + 0.577·3-s + (0.249 − 0.433i)4-s + (0.214 − 0.123i)5-s + (0.353 − 0.204i)6-s + (0.837 − 0.546i)7-s − 0.353i·8-s + 0.333·9-s + (0.0875 − 0.151i)10-s + (−0.246 − 0.426i)11-s + (0.144 − 0.249i)12-s + (−0.103 − 0.178i)13-s + (0.319 − 0.630i)14-s + (0.123 − 0.0715i)15-s + (−0.125 − 0.216i)16-s − 0.377i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.482 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49578 - 1.47426i\)
\(L(\frac12)\) \(\approx\) \(2.49578 - 1.47426i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.21 + 1.44i)T \)
19 \( 1 + (3.86 - 2.00i)T \)
good5 \( 1 + (-0.479 + 0.276i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.816 + 1.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.372 + 0.645i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.55iT - 17T^{2} \)
23 \( 1 - 4.40T + 23T^{2} \)
29 \( 1 + (-1.84 + 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 0.891i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.12 + 1.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.16 - 3.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.32iT - 47T^{2} \)
53 \( 1 + (-1.51 - 0.873i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.11T + 59T^{2} \)
61 \( 1 - 4.19iT - 61T^{2} \)
67 \( 1 + (-2.98 - 1.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.47 + 0.850i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.87iT - 73T^{2} \)
79 \( 1 + (-2.49 + 1.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.44iT - 83T^{2} \)
89 \( 1 - 6.81T + 89T^{2} \)
97 \( 1 + (6.71 - 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.31457947594850830738334065141, −9.326314329505451790801070989844, −8.397062884372410668485284986219, −7.60572070899116865610550100728, −6.62568162767101421644570863748, −5.44879696018999331511530542132, −4.62494858082863255446619268360, −3.64502142652816572351037920070, −2.52690480850524485998359062122, −1.29121306707482559800993056395, 1.93175956746451952403980983394, 2.81948834399051886473714973449, 4.22252790340877515209302824433, 4.92323084134571692217708451549, 6.01142001756457853377206125109, 6.92347602170953441925114829750, 7.892059228835910023405987119235, 8.539172145411466400960096042553, 9.407075828566091349917443217264, 10.49014550048886922518050176936

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