↑

Properties

Label 2-847-11.5-c1-0-12
Degree $2$
Conductor $847$
Sign $0.923 + 0.382i$
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  + (−0.0371 − 0.114i)2-s + (−2.23 − 1.62i)3-s + (1.60 − 1.16i)4-s + (−0.867 + 2.67i)5-s + (−0.102 + 0.316i)6-s + (0.809 − 0.587i)7-s + (−0.388 − 0.281i)8-s + (1.44 + 4.43i)9-s + 0.338·10-s − 5.49·12-s + (−0.333 − 1.02i)13-s + (−0.0973 − 0.0707i)14-s + (6.28 − 4.56i)15-s + (1.20 − 3.72i)16-s + (−2.15 + 6.61i)17-s + (0.453 − 0.329i)18-s + ⋯
L(s)  = 1  + (−0.0262 − 0.0809i)2-s + (−1.29 − 0.939i)3-s + (0.803 − 0.583i)4-s + (−0.388 + 1.19i)5-s + (−0.0420 + 0.129i)6-s + (0.305 − 0.222i)7-s + (−0.137 − 0.0996i)8-s + (0.480 + 1.47i)9-s + 0.106·10-s − 1.58·12-s + (−0.0925 − 0.284i)13-s + (−0.0260 − 0.0189i)14-s + (1.62 − 1.17i)15-s + (0.302 − 0.930i)16-s + (−0.521 + 1.60i)17-s + (0.106 − 0.0777i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11719 - 0.222177i\)
\(L(\frac12)\) \(\approx\) \(1.11719 - 0.222177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.0371 + 0.114i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.23 + 1.62i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.867 - 2.67i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.333 + 1.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.15 - 6.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.10 - 4.43i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + (-0.992 + 0.721i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.49 + 7.67i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.24 - 0.902i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.52 - 5.46i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (-1.53 - 1.11i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.18 + 3.63i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.39 + 3.91i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.02 + 9.31i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 + (-3.86 + 11.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.07 + 1.50i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.73 - 14.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.632 - 1.94i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + (-2.81 - 8.66i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.61809040315657380249791108332, −9.682253439131098800747488116811, −7.86305208531297039327807992879, −7.43293800422876382595116860977, −6.53128674250863100052105108712, −6.07663235926956675720980589307, −5.19540209380006773114126086387, −3.58853668220551921366807962810, −2.22079108695152799369012961018, −1.04751504196804686639048613630, 0.860667858456044097702914105915, 2.84845858577610937107843043128, 4.21716601431202543377160818510, 5.01517788273063676536042820397, 5.50860004355981852259952553214, 6.87060400453050489558472031810, 7.46176193841504670847327548554, 8.911793098715005869795122175491, 9.182587010693652289089257402579, 10.48872824547145740251496006443

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