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Properties

Label 2-847-11.9-c1-0-40
Degree $2$
Conductor $847$
Sign $0.780 + 0.625i$
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  + (2.42 − 1.76i)3-s + (1.61 + 1.17i)4-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (1.85 − 5.70i)9-s + 6·12-s + (−1.23 + 3.80i)13-s + (−2.42 − 1.76i)15-s + (1.23 + 3.80i)16-s + (0.618 + 1.90i)17-s + (4.85 − 3.52i)19-s + (0.618 − 1.90i)20-s + 3·21-s − 5·23-s + (3.23 − 2.35i)25-s + ⋯
L(s)  = 1  + (1.40 − 1.01i)3-s + (0.809 + 0.587i)4-s + (−0.138 − 0.425i)5-s + (0.305 + 0.222i)7-s + (0.618 − 1.90i)9-s + 1.73·12-s + (−0.342 + 1.05i)13-s + (−0.626 − 0.455i)15-s + (0.309 + 0.951i)16-s + (0.149 + 0.461i)17-s + (1.11 − 0.809i)19-s + (0.138 − 0.425i)20-s + 0.654·21-s − 1.04·23-s + (0.647 − 0.470i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.780 + 0.625i$
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: \((2,\ 847,\ (\ :1/2),\ 0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.80132 - 0.984421i\)
\(L(\frac12)\) \(\approx\) \(2.80132 - 0.984421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.42 + 1.76i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.23 - 3.80i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.618 - 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.85 + 3.52i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 + (8.09 + 5.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.04 - 2.93i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.61 + 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.85 - 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.42 + 1.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.618 + 1.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.09 + 5.87i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.85 + 5.70i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.70 - 11.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (1.54 - 4.75i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.753671676855240177644420885806, −9.040310955317030568855175922687, −8.173538262825589658740771463950, −7.70674897683723552635041874944, −6.95619799680519449721577433088, −6.08442812907609159812148395059, −4.42618131772671582762030967788, −3.34293613773245273288453843195, −2.37437655398428988609208952775, −1.56563894758485742161374946476, 1.76716494670071560451818519596, 2.99348622704762871006537909329, 3.52164880763805627811646996152, 4.91111853859233805660516725604, 5.71489902108833425034507148382, 7.23650483107099833224299975203, 7.68375035911538302908679623572, 8.617208031956212201983764200266, 9.812558609856527449166731097618, 9.980837963824698150779298666508

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