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Properties

Label 2-847-11.5-c1-0-17
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  + (0.835 + 2.57i)2-s + (2.27 + 1.65i)3-s + (−4.29 + 3.11i)4-s + (−0.137 + 0.423i)5-s + (−2.34 + 7.21i)6-s + (−0.809 + 0.587i)7-s + (−7.23 − 5.25i)8-s + (1.50 + 4.64i)9-s − 1.20·10-s − 14.8·12-s + (0.139 + 0.428i)13-s + (−2.18 − 1.58i)14-s + (−1.01 + 0.734i)15-s + (4.18 − 12.8i)16-s + (1.49 − 4.59i)17-s + (−10.6 + 7.75i)18-s + ⋯
L(s)  = 1  + (0.590 + 1.81i)2-s + (1.31 + 0.952i)3-s + (−2.14 + 1.55i)4-s + (−0.0615 + 0.189i)5-s + (−0.957 + 2.94i)6-s + (−0.305 + 0.222i)7-s + (−2.55 − 1.85i)8-s + (0.502 + 1.54i)9-s − 0.380·10-s − 4.30·12-s + (0.0386 + 0.118i)13-s + (−0.584 − 0.424i)14-s + (−0.260 + 0.189i)15-s + (1.04 − 3.22i)16-s + (0.362 − 1.11i)17-s + (−2.51 + 1.82i)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: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15517 - 2.30422i\)
\(L(\frac12)\) \(\approx\) \(1.15517 - 2.30422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.835 - 2.57i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.27 - 1.65i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.137 - 0.423i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.139 - 0.428i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.49 + 4.59i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.878 - 0.638i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + (-1.60 + 1.16i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.55 - 7.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.91 - 4.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.43 + 1.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (0.827 + 0.601i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.10 + 3.39i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.6 + 8.46i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.52 + 4.68i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + (1.82 - 5.63i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.34 + 0.974i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.11 + 3.43i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.33 - 10.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + (1.64 + 5.04i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.19096190734535835691481371021, −9.495168301959178454150360914054, −8.709852494146047287787610000316, −8.306485856232465548622896848317, −7.20524711427234861857589123419, −6.66573410221180617047615676980, −5.18774808022348936098314105321, −4.80878032348561933420658587255, −3.49964646453680712877189832651, −3.06532944635825486957123055196, 0.966530405543048866706304738520, 1.98690623295309967927681532306, 2.96495252397256421947232611231, 3.62261831285094392991683585945, 4.67788913768881929322357712944, 5.96816725187249990499135617195, 7.18992815355604167323732823256, 8.441110238332143647333223488907, 8.817233901097653360828439287096, 9.835350350087401381583936915253

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