↑

Properties

Label 2-847-11.3-c1-0-35
Degree $2$
Conductor $847$
Sign $0.898 + 0.437i$
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  + (−1.70 + 1.23i)2-s + (0.524 + 1.61i)3-s + (0.756 − 2.32i)4-s + (−0.398 − 0.289i)5-s + (−2.89 − 2.10i)6-s + (0.309 − 0.951i)7-s + (0.293 + 0.901i)8-s + (0.0930 − 0.0676i)9-s + 1.03·10-s + 4.16·12-s + (−4.28 + 3.11i)13-s + (0.651 + 2.00i)14-s + (0.258 − 0.795i)15-s + (2.34 + 1.70i)16-s + (−2.45 − 1.78i)17-s + (−0.0749 + 0.230i)18-s + ⋯
L(s)  = 1  + (−1.20 + 0.876i)2-s + (0.303 + 0.932i)3-s + (0.378 − 1.16i)4-s + (−0.178 − 0.129i)5-s + (−1.18 − 0.859i)6-s + (0.116 − 0.359i)7-s + (0.103 + 0.318i)8-s + (0.0310 − 0.0225i)9-s + 0.328·10-s + 1.20·12-s + (−1.18 + 0.864i)13-s + (0.174 + 0.536i)14-s + (0.0667 − 0.205i)15-s + (0.586 + 0.425i)16-s + (−0.595 − 0.432i)17-s + (−0.0176 + 0.0544i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.898 + 0.437i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.898 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428466 - 0.0988225i\)
\(L(\frac12)\) \(\approx\) \(0.428466 - 0.0988225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (1.70 - 1.23i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.524 - 1.61i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.398 + 0.289i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.28 - 3.11i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.45 + 1.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.44 + 4.43i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 + (-2.13 + 6.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.02 + 0.743i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.36 + 10.3i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.445 + 1.37i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.88T + 43T^{2} \)
47 \( 1 + (2.70 + 8.32i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.37 - 3.90i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.58 - 7.94i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.2 - 8.16i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 + (4.81 + 3.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.16 - 3.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.12 + 5.17i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.97 + 6.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (-5.11 + 3.71i)T + (29.9 - 92.2i)T^{2} \)
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

−9.899590490687609412560759830402, −9.214749468809408140578824816903, −8.636254373975314459140418993265, −7.60120357582580550099926426415, −7.00412453263633450877342013475, −6.04787927288462949756627707127, −4.59598850290062710821088225079, −4.07491306301090151416713022325, −2.34031416834008257485282098719, −0.31127232037346684740167497818, 1.41457670493056793740142792058, 2.26396511616123970845418147210, 3.24201101967098207258616541088, 4.86199844227171595166722631388, 6.14633624528896972027275137711, 7.25941306767695324985059431199, 8.071927972466530976509510888063, 8.334490094053789071231340179918, 9.587112922967866162038377608752, 10.12958995144092845030193511134

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