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Properties

Label 2-847-11.9-c1-0-5
Degree $2$
Conductor $847$
Sign $-0.598 - 0.801i$
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  + (0.0371 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.598 - 0.801i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319928 + 0.638161i\)
\(L(\frac12)\) \(\approx\) \(0.319928 + 0.638161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.81 + 8.66i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.58002091193418126875103707142, −10.01803504224514286573714053476, −8.663940742885480300874446723229, −8.152765623512524325044994343051, −6.81340668911729555218168458259, −6.03496279319360457770233699560, −5.15904298893972856118440850988, −4.17746746079088058963933633154, −3.51024252523116504955261695588, −1.43716833067234088074468015687, 0.40874467222965471284874190785, 2.05297876182843590271506407687, 3.07548069120367606389239440101, 4.89391881091216811674655305112, 5.76383700673542172646562738920, 6.72228797188972980805891342041, 6.88554710754889231912974108870, 7.61917479656950142425547146374, 9.177286543111673911152501845018, 10.33991930353680700131944675282

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