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Properties

Label 2-798-133.103-c1-0-3
Degree $2$
Conductor $798$
Sign $-0.641 - 0.766i$
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 + (−2.99 + 1.73i)5-s + (−0.866 + 0.5i)6-s + (2.47 + 0.942i)7-s + 0.999i·8-s + 9-s + (1.73 − 2.99i)10-s + (−0.833 − 1.44i)11-s + (0.499 − 0.866i)12-s + (0.376 + 0.652i)13-s + (−2.61 + 0.420i)14-s + (−2.99 + 1.73i)15-s + (−0.5 − 0.866i)16-s + 2.71i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.577·3-s + (0.249 − 0.433i)4-s + (−1.34 + 0.774i)5-s + (−0.353 + 0.204i)6-s + (0.934 + 0.356i)7-s + 0.353i·8-s + 0.333·9-s + (0.547 − 0.948i)10-s + (−0.251 − 0.435i)11-s + (0.144 − 0.249i)12-s + (0.104 + 0.181i)13-s + (−0.698 + 0.112i)14-s + (−0.774 + 0.446i)15-s + (−0.125 − 0.216i)16-s + 0.658i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.641 - 0.766i$
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: \((2,\ 798,\ (\ :1/2),\ -0.641 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386923 + 0.828540i\)
\(L(\frac12)\) \(\approx\) \(0.386923 + 0.828540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.47 - 0.942i)T \)
19 \( 1 + (-0.186 - 4.35i)T \)
good5 \( 1 + (2.99 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.833 + 1.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.376 - 0.652i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.71iT - 17T^{2} \)
23 \( 1 - 0.179T + 23T^{2} \)
29 \( 1 + (6.09 - 3.51i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.13 - 1.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.60 + 2.65i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.08 - 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.07 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.39iT - 47T^{2} \)
53 \( 1 + (7.18 + 4.15i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.43T + 59T^{2} \)
61 \( 1 + 0.172iT - 61T^{2} \)
67 \( 1 + (-0.393 - 0.227i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.60 - 5.54i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.04iT - 73T^{2} \)
79 \( 1 + (12.9 - 7.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.65iT - 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (6.55 - 11.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68054303917818377805517945955, −9.631328140810035014621092696656, −8.438607782508004533963540889518, −8.164157033137183356743400571744, −7.43612099702189234829691919219, −6.52357662087338845447477300929, −5.30159625195645343613396033098, −4.04554892034847411299590359417, −3.12707188592397057007591136580, −1.67907898035842443775933502850, 0.53085238448717571873379517631, 2.00331157216886730438499289675, 3.43682903821270591881793791154, 4.36261993722451102859847326004, 5.14342674349524859997113623302, 7.10465576948848601657228355759, 7.56991917881774392161722900959, 8.370264721931869325034310602353, 8.880505599506503510064157264946, 9.877596466480983164957545456392

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