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Properties

Label 2-847-77.62-c1-0-43
Degree $2$
Conductor $847$
Sign $0.307 + 0.951i$
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  + (0.912 − 1.25i)2-s + (−0.126 − 0.390i)4-s + (2.51 − 0.817i)7-s + (2.34 + 0.762i)8-s + (−2.42 − 1.76i)9-s + (1.26 − 3.90i)14-s + (3.76 − 2.73i)16-s + (−4.43 + 1.43i)18-s + 9.58·23-s + (1.54 − 4.75i)25-s + (−0.638 − 0.879i)28-s + (−1.30 + 0.422i)29-s − 2.28i·32-s + (−0.380 + 1.17i)36-s + (−0.422 − 1.29i)37-s + ⋯
L(s)  = 1  + (0.645 − 0.888i)2-s + (−0.0634 − 0.195i)4-s + (0.951 − 0.309i)7-s + (0.829 + 0.269i)8-s + (−0.809 − 0.587i)9-s + (0.339 − 1.04i)14-s + (0.941 − 0.683i)16-s + (−1.04 + 0.339i)18-s + 1.99·23-s + (0.309 − 0.951i)25-s + (−0.120 − 0.166i)28-s + (−0.241 + 0.0785i)29-s − 0.404i·32-s + (−0.0634 + 0.195i)36-s + (−0.0694 − 0.213i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.307 + 0.951i$
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: \((2,\ 847,\ (\ :1/2),\ 0.307 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03740 - 1.48304i\)
\(L(\frac12)\) \(\approx\) \(2.03740 - 1.48304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.51 + 0.817i)T \)
11 \( 1 \)
good2 \( 1 + (-0.912 + 1.25i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 9.58T + 23T^{2} \)
29 \( 1 + (1.30 - 0.422i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.422 + 1.29i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.77iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.6 + 7.73i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 16.3T + 67T^{2} \)
71 \( 1 + (-8.07 + 5.86i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.35 - 10.1i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.38202799333719768855115564002, −9.218940130187860289012312101201, −8.354871820643901460932828551701, −7.54810910660171163847206739479, −6.48566656145350457332690017901, −5.21305989004111770576778193243, −4.55520206863476892456217653129, −3.44475404240616735478826661865, −2.58084371224135169225030209750, −1.20788482193988920509430622694, 1.55036540918887681457544835324, 3.00050566460606238052615302895, 4.47351168926182201771166930396, 5.21957410190387835966078754042, 5.74464634631672774433235561800, 6.93188456967400770688470344106, 7.61042456745548176418083745842, 8.489669217762675968344616460786, 9.276016989894749290392973550427, 10.71102128769441791241362268669

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