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Properties

Label 2-847-11.4-c1-0-48
Degree $2$
Conductor $847$
Sign $0.859 + 0.511i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 + 1.31i)2-s + (1 − 3.07i)3-s + (0.927 + 2.85i)4-s + (1.61 − 1.17i)5-s + (5.85 − 4.25i)6-s + (0.309 + 0.951i)7-s + (−0.690 + 2.12i)8-s + (−6.04 − 4.39i)9-s + 4.47·10-s + 9.70·12-s + (1 + 0.726i)13-s + (−0.690 + 2.12i)14-s + (−2 − 6.15i)15-s + (0.809 − 0.587i)16-s + (−1 + 0.726i)17-s + (−5.16 − 15.8i)18-s + ⋯
L(s)  = 1  + (1.27 + 0.929i)2-s + (0.577 − 1.77i)3-s + (0.463 + 1.42i)4-s + (0.723 − 0.525i)5-s + (2.38 − 1.73i)6-s + (0.116 + 0.359i)7-s + (−0.244 + 0.751i)8-s + (−2.01 − 1.46i)9-s + 1.41·10-s + 2.80·12-s + (0.277 + 0.201i)13-s + (−0.184 + 0.568i)14-s + (−0.516 − 1.58i)15-s + (0.202 − 0.146i)16-s + (−0.242 + 0.176i)17-s + (−1.21 − 3.74i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.85372 - 1.05991i\)
\(L(\frac12)\) \(\approx\) \(3.85372 - 1.05991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 1.31i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1 + 3.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.61 + 1.17i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1 - 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1 - 0.726i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.763 - 2.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (0.145 + 0.449i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.85 - 4.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.145 - 0.449i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.09 - 6.43i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-2.23 + 6.88i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.85 + 4.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1 - 3.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.23 + 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + (-1.23 + 0.898i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.61 + 4.97i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.23 + 5.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.4 - 9.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (-7.61 - 5.53i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.911135255306864696984302164111, −8.762157095015476349165047193501, −8.131366297719000829202924100429, −7.37443066668204771055948246133, −6.34353222461146246793822244591, −6.09187149895638928439083289935, −5.10997216003410429566066280035, −3.75252734226965102979045515247, −2.51829477323063645667669838233, −1.45199124256636642421017900680, 2.26447983737144466092442076870, 2.94265753153118761599287123224, 4.00395213354098157455633564587, 4.45709040681305809395426708694, 5.49126159159543834292520693756, 6.21114604620390160060222242133, 7.917474883960857013233409058646, 8.949060194649923266063115991027, 9.944072674441028186303383363567, 10.27870125994927796207820081236

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