↑

Properties

Label 2-847-7.4-c1-0-59
Degree $2$
Conductor $847$
Sign $-0.910 + 0.414i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0380 + 0.0658i)2-s + (1.17 − 2.03i)3-s + (0.997 − 1.72i)4-s + (−1.28 − 2.23i)5-s + 0.179·6-s + (−1.24 − 2.33i)7-s + 0.303·8-s + (−1.27 − 2.20i)9-s + (0.0979 − 0.169i)10-s + (−2.34 − 4.06i)12-s + 4.59·13-s + (0.106 − 0.170i)14-s − 6.07·15-s + (−1.98 − 3.43i)16-s + (−1.27 + 2.20i)17-s + (0.0967 − 0.167i)18-s + ⋯
L(s)  = 1  + (0.0268 + 0.0465i)2-s + (0.679 − 1.17i)3-s + (0.498 − 0.863i)4-s + (−0.576 − 0.998i)5-s + 0.0730·6-s + (−0.470 − 0.882i)7-s + 0.107·8-s + (−0.424 − 0.735i)9-s + (0.0309 − 0.0536i)10-s + (−0.677 − 1.17i)12-s + 1.27·13-s + (0.0284 − 0.0456i)14-s − 1.56·15-s + (−0.495 − 0.858i)16-s + (−0.308 + 0.534i)17-s + (0.0228 − 0.0395i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.910 + 0.414i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (606, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.910 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420247 - 1.93899i\)
\(L(\frac12)\) \(\approx\) \(0.420247 - 1.93899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.24 + 2.33i)T \)
11 \( 1 \)
good2 \( 1 + (-0.0380 - 0.0658i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.17 + 2.03i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.28 + 2.23i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.59T + 13T^{2} \)
17 \( 1 + (1.27 - 2.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.31 - 5.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.10T + 29T^{2} \)
31 \( 1 + (-1.88 + 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.359T + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 + (-3.14 - 5.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.22 - 3.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.75 - 3.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.29 - 3.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 + (3.28 - 5.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.24 + 9.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.66T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.748911054270061272159687190185, −8.809561932812167104291672136687, −7.998861318039215151190187927082, −7.34072979956136282942739499114, −6.47996552145501396788171605827, −5.64293333202871649325646613955, −4.30480179279451480412817378237, −3.22584859270700124251089739382, −1.59648662381179822014023260430, −0.971941492150407549213974488372, 2.65410200700901619378382964333, 3.12440010999168121230288157865, 3.86051576468735241554828106196, 4.99906317067664786952260338090, 6.54150410025122191422946635867, 7.00418302216262858031281571845, 8.386562004056271793294904924878, 8.709007687981204301009664232007, 9.654711871160293308956392017592, 10.68201375522166373541293173655

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