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Properties

Label 2-798-57.50-c1-0-25
Degree $2$
Conductor $798$
Sign $0.819 + 0.573i$
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  + (−0.5 − 0.866i)2-s + (1.69 + 0.343i)3-s + (−0.499 + 0.866i)4-s + (1.60 − 0.924i)5-s + (−0.551 − 1.64i)6-s − 7-s + 0.999·8-s + (2.76 + 1.16i)9-s + (−1.60 − 0.924i)10-s + 2.59i·11-s + (−1.14 + 1.29i)12-s + (−1.08 − 0.624i)13-s + (0.5 + 0.866i)14-s + (3.03 − 1.01i)15-s + (−0.5 − 0.866i)16-s + (4.74 − 2.73i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.980 + 0.198i)3-s + (−0.249 + 0.433i)4-s + (0.716 − 0.413i)5-s + (−0.225 − 0.670i)6-s − 0.377·7-s + 0.353·8-s + (0.921 + 0.388i)9-s + (−0.506 − 0.292i)10-s + 0.783i·11-s + (−0.330 + 0.374i)12-s + (−0.300 − 0.173i)13-s + (0.133 + 0.231i)14-s + (0.784 − 0.263i)15-s + (−0.125 − 0.216i)16-s + (1.15 − 0.664i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92244 - 0.605966i\)
\(L(\frac12)\) \(\approx\) \(1.92244 - 0.605966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.69 - 0.343i)T \)
7 \( 1 + T \)
19 \( 1 + (-1.34 + 4.14i)T \)
good5 \( 1 + (-1.60 + 0.924i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.59iT - 11T^{2} \)
13 \( 1 + (1.08 + 0.624i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.74 + 2.73i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.68 - 3.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.51 - 2.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.56iT - 31T^{2} \)
37 \( 1 - 1.37iT - 37T^{2} \)
41 \( 1 + (5.42 + 9.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.656 + 1.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.40 - 4.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.07 - 7.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.000349 + 0.000605i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.95 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.72 - 0.997i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.534 - 0.926i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.59 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.27 + 3.62i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (-0.493 + 0.854i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.1 - 7.00i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.946892640448003106275188755041, −9.251473690293234194008500689414, −9.035763041319935190694245416293, −7.56672217102139682456266696807, −7.21739079653923098133439278595, −5.51794291405600216630886575210, −4.64704283966465524563904475590, −3.38155716237483819548427402849, −2.53571510556418690202049382107, −1.32904032725694134035732470357, 1.37139956803334186575614331574, 2.77560131627940902167730572967, 3.72451660627416788881694837105, 5.21604926379370017633312274942, 6.24881721600181839369937005173, 6.87802534360495451999132050730, 7.944574060287720814558156956162, 8.514890965444749170480238371588, 9.501746249363665737301731350566, 10.02937601336420200930960623696

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