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Properties

Label 2-798-7.4-c1-0-20
Degree $2$
Conductor $798$
Sign $0.291 + 0.956i$
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 + (−1.24 − 2.15i)5-s + 0.999·6-s + (2.47 − 0.929i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.24 − 2.15i)10-s + (1.73 − 2.99i)11-s + (0.499 + 0.866i)12-s − 4.98·13-s + (2.04 + 1.68i)14-s − 2.49·15-s + (−0.5 − 0.866i)16-s + (−1.12 + 1.94i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.557 − 0.965i)5-s + 0.408·6-s + (0.936 − 0.351i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.394 − 0.682i)10-s + (0.521 − 0.903i)11-s + (0.144 + 0.249i)12-s − 1.38·13-s + (0.546 + 0.449i)14-s − 0.643·15-s + (−0.125 − 0.216i)16-s + (−0.272 + 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30985 - 0.970475i\)
\(L(\frac12)\) \(\approx\) \(1.30985 - 0.970475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.47 + 0.929i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 + 2.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.98T + 13T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.00 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + (-3.64 + 6.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.58 - 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 - 9.54T + 43T^{2} \)
47 \( 1 + (2.68 + 4.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.93 + 5.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.33 + 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.18 + 3.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.57 - 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + (7.21 - 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.54 - 6.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.59T + 83T^{2} \)
89 \( 1 + (3.76 + 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.00T + 97T^{2} \)
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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

−9.913856301923551882143402774145, −8.837436259394706769851973924131, −8.220651692928996736194043636240, −7.69837552029603347002841277220, −6.69269134457663136743511192300, −5.64995413672672967147243563885, −4.59531648617437197206587770056, −4.01681839210996036307627232106, −2.37704828708127324626432141570, −0.70217313964319743459085512356, 1.97032111655133572341905052556, 2.88288993892061144680541578947, 4.07649851679416914230629760012, 4.74687970818270516791400085736, 5.81304363847087432844234914459, 7.29678992859357501474182914781, 7.64556222266778381876118309330, 9.087058144381190762064201026605, 9.636242896508000792370884205760, 10.61868503713108248958395180511

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