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Properties

Label 2-847-77.62-c1-0-21
Degree $2$
Conductor $847$
Sign $0.800 - 0.599i$
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  + (0.513 − 0.707i)2-s + (1.26 + 0.410i)3-s + (0.381 + 1.17i)4-s + (−0.780 − 1.07i)5-s + (0.939 − 0.682i)6-s + (−0.169 + 2.64i)7-s + (2.68 + 0.874i)8-s + (−0.999 − 0.726i)9-s − 1.16·10-s + 1.64i·12-s + (4.91 + 3.57i)13-s + (1.77 + 1.47i)14-s + (−0.545 − 1.67i)15-s + (−3.39 + 2.46i)17-s + (−1.02 + 0.333i)18-s + (−0.939 + 2.89i)19-s + ⋯
L(s)  = 1  + (0.363 − 0.499i)2-s + (0.729 + 0.236i)3-s + (0.190 + 0.587i)4-s + (−0.349 − 0.480i)5-s + (0.383 − 0.278i)6-s + (−0.0641 + 0.997i)7-s + (0.951 + 0.309i)8-s + (−0.333 − 0.242i)9-s − 0.367·10-s + 0.473i·12-s + (1.36 + 0.990i)13-s + (0.475 + 0.394i)14-s + (−0.140 − 0.433i)15-s + (−0.824 + 0.598i)17-s + (−0.242 + 0.0786i)18-s + (−0.215 + 0.663i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28109 + 0.758820i\)
\(L(\frac12)\) \(\approx\) \(2.28109 + 0.758820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.169 - 2.64i)T \)
11 \( 1 \)
good2 \( 1 + (-0.513 + 0.707i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.26 - 0.410i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.780 + 1.07i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.91 - 3.57i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.39 - 2.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.939 - 2.89i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-6.52 + 2.12i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.30 - 4.55i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.927 + 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.939 - 2.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.61iT - 43T^{2} \)
47 \( 1 + (-9.14 - 2.96i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-6.91 + 2.24i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.45 + 1.78i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (0.809 - 0.587i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.09 + 6.46i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.49 + 3.43i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.01 - 5.09i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 18.5iT - 89T^{2} \)
97 \( 1 + (3.30 - 4.55i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.44072170301463880363011415822, −8.973578491767362353088069061225, −8.735099619896502319308755986634, −8.151473433622200228550101583539, −6.80498627722931252129835361165, −5.87730553295522230214589677569, −4.49059364947667084930902852628, −3.80710124804519557142996743233, −2.86743343599349336288601334555, −1.82859326544296834606246369919, 1.03726206510059759214087888523, 2.65812231094512639241693690811, 3.66689973731201271729524880498, 4.77499584082406168431366260086, 5.81030011200834335106859615989, 6.85540109301688134960469889497, 7.32144083508173355351433830125, 8.263497735206417413781678707895, 9.099005062589329648941725710678, 10.29669040363713655922063918234

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