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Properties

Label 2-798-7.2-c1-0-8
Degree $2$
Conductor $798$
Sign $-0.198 - 0.980i$
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 + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.792 − 1.37i)5-s − 0.999·6-s + (−2.62 + 0.358i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.792 + 1.37i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + 2.82·13-s + (1 − 2.44i)14-s + 1.58·15-s + (−0.5 + 0.866i)16-s + (2.12 + 3.67i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.354 − 0.614i)5-s − 0.408·6-s + (−0.990 + 0.135i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.250 + 0.434i)10-s + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s + 0.784·13-s + (0.267 − 0.654i)14-s + 0.409·15-s + (−0.125 + 0.216i)16-s + (0.514 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802749 + 0.981178i\)
\(L(\frac12)\) \(\approx\) \(0.802749 + 0.981178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.792 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.70 - 2.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + (-5.32 - 9.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 + (1.58 - 2.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.914 + 1.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.03 + 5.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.41 + 4.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.89T + 71T^{2} \)
73 \( 1 + (-7.41 - 12.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0857 + 0.148i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + (1.87 - 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.58T + 97T^{2} \)
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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

−10.12300005515334745799390077874, −9.631213843771337371779962372232, −8.764135182241552881162159478741, −8.238328008119692624583709414550, −7.00271614096463559887752895961, −6.10600809800832053534029359640, −5.36796568687533943277449717096, −4.20097969927766368715433452917, −3.16270185061131601814143916962, −1.41296649158129671546084947271, 0.75892320862623553526495040214, 2.43503362928647196856039301507, 3.13088137543958099007457933335, 4.24211277218324257306097834592, 5.92300055147662346052161628481, 6.57089355671868198704372718759, 7.49667934836725759437598019541, 8.457487669278599402285477276248, 9.312539569981503718532147208786, 10.02918356727781457107830763543

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