↑

Properties

Label 2-847-7.4-c1-0-19
Degree $2$
Conductor $847$
Sign $-0.694 - 0.719i$
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  + (0.328 + 0.568i)2-s + (−0.956 + 1.65i)3-s + (0.784 − 1.35i)4-s + (1.78 + 3.09i)5-s − 1.25·6-s + (−1.78 + 1.95i)7-s + 2.34·8-s + (−0.328 − 0.568i)9-s + (−1.17 + 2.02i)10-s + (1.5 + 2.59i)12-s + 5.91·13-s + (−1.69 − 0.373i)14-s − 6.82·15-s + (−0.799 − 1.38i)16-s + (−0.828 + 1.43i)17-s + (0.215 − 0.373i)18-s + ⋯
L(s)  = 1  + (0.232 + 0.402i)2-s + (−0.552 + 0.956i)3-s + (0.392 − 0.679i)4-s + (0.798 + 1.38i)5-s − 0.512·6-s + (−0.674 + 0.738i)7-s + 0.828·8-s + (−0.109 − 0.189i)9-s + (−0.370 + 0.641i)10-s + (0.433 + 0.749i)12-s + 1.63·13-s + (−0.453 − 0.0997i)14-s − 1.76·15-s + (−0.199 − 0.346i)16-s + (−0.200 + 0.347i)17-s + (0.0508 − 0.0880i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.694 - 0.719i$
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: \((2,\ 847,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723354 + 1.70235i\)
\(L(\frac12)\) \(\approx\) \(0.723354 + 1.70235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.78 - 1.95i)T \)
11 \( 1 \)
good2 \( 1 + (-0.328 - 0.568i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.956 - 1.65i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.78 - 3.09i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + (0.828 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.740 - 1.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.08T + 29T^{2} \)
31 \( 1 + (-3.54 + 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 + 1.59T + 43T^{2} \)
47 \( 1 + (-0.828 - 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.61 - 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.42 - 7.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.61T + 71T^{2} \)
73 \( 1 + (-2.28 + 3.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + (-1.28 - 2.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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

−10.54443784298690700361749277281, −9.902314656224592409589524913101, −9.182748800542806488060515185263, −7.77719528697437101207292744754, −6.48053931817174017781107546903, −6.13691006742294737033776502937, −5.62365917491693103381276562574, −4.32932186710055014215906136480, −3.13956689022742808748953726882, −1.93904696950105151237422321996, 0.952250624242046489166246695934, 1.77799648595026691359115020179, 3.38898270654476440397778750795, 4.36352386491777842390129760283, 5.61372030112510088258822527475, 6.41538059243465944397504147015, 7.16373844746366004727687543844, 8.151470910693996208223761757030, 8.997808819085119596191440028991, 9.904497450588895188562447691272

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