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Properties

Label 2-847-77.13-c1-0-62
Degree $2$
Conductor $847$
Sign $-0.997 - 0.0771i$
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  + (1.16 + 0.379i)2-s + (−0.767 − 1.05i)3-s + (−0.395 − 0.287i)4-s + (−2.26 + 0.737i)5-s + (−0.496 − 1.52i)6-s + (2.39 + 1.12i)7-s + (−1.79 − 2.47i)8-s + (0.400 − 1.23i)9-s − 2.93·10-s + 0.638i·12-s + (−0.802 + 2.47i)13-s + (2.37 + 2.22i)14-s + (2.52 + 1.83i)15-s + (−0.860 − 2.64i)16-s + (−1.40 − 4.31i)17-s + (0.935 − 1.28i)18-s + ⋯
L(s)  = 1  + (0.826 + 0.268i)2-s + (−0.443 − 0.609i)3-s + (−0.197 − 0.143i)4-s + (−1.01 + 0.329i)5-s + (−0.202 − 0.623i)6-s + (0.905 + 0.423i)7-s + (−0.635 − 0.875i)8-s + (0.133 − 0.410i)9-s − 0.927·10-s + 0.184i·12-s + (−0.222 + 0.685i)13-s + (0.635 + 0.593i)14-s + (0.650 + 0.472i)15-s + (−0.215 − 0.661i)16-s + (−0.340 − 1.04i)17-s + (0.220 − 0.303i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.997 - 0.0771i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.997 - 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00760279 + 0.196830i\)
\(L(\frac12)\) \(\approx\) \(0.00760279 + 0.196830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.39 - 1.12i)T \)
11 \( 1 \)
good2 \( 1 + (-1.16 - 0.379i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.767 + 1.05i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (2.26 - 0.737i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.802 - 2.47i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.40 + 4.31i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.65 - 4.83i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 + (1.65 - 2.28i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.93 + 1.27i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.25 + 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.654 + 0.475i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (1.08 + 1.48i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.773 - 2.38i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.33 + 8.71i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.91 + 5.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (3.05 + 9.40i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.87 + 2.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.58 + 1.49i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.20 - 9.86i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.52iT - 89T^{2} \)
97 \( 1 + (9.72 + 3.16i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.707508999372165649855247998723, −8.811834735662715718153717136863, −7.82800416763272437403339443125, −6.98777390646521926753162043302, −6.24260100204341416061741236284, −5.32280968220260235142995929993, −4.32376585087765056237256648091, −3.66185043837485790589421622258, −1.90507397215152713811293824054, −0.07503434267870270872054772316, 2.22136845932134680118646815808, 3.85184010984863954096754416155, 4.28770429736359126222711348775, 4.94667592542520426379262395649, 5.85590679199400505407280183311, 7.32580519076741110102706893493, 8.272381476567055719118406508573, 8.604709375218703881657998421615, 10.10315477923927062731311121052, 10.91615350634590919831571325533

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