Properties

Label 2-847-77.76-c1-0-3
Degree $2$
Conductor $847$
Sign $0.0681 - 0.997i$
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  − 1.93i·2-s − 1.73·4-s + 3.31i·5-s + (−2.34 + 1.22i)7-s − 0.517i·8-s + 3·9-s + 6.40·10-s − 6.40·13-s + (2.36 + 4.53i)14-s − 4.46·16-s − 1.71·17-s − 5.79i·18-s − 4.69·19-s − 5.74i·20-s − 4.73·23-s + ⋯
L(s)  = 1  − 1.36i·2-s − 0.866·4-s + 1.48i·5-s + (−0.886 + 0.462i)7-s − 0.183i·8-s + 9-s + 2.02·10-s − 1.77·13-s + (0.632 + 1.21i)14-s − 1.11·16-s − 0.416·17-s − 1.36i·18-s − 1.07·19-s − 1.28i·20-s − 0.986·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0681 - 0.997i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.0681 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308223 + 0.287890i\)
\(L(\frac12)\) \(\approx\) \(0.308223 + 0.287890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.34 - 1.22i)T \)
11 \( 1 \)
good2 \( 1 + 1.93iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
5 \( 1 - 3.31iT - 5T^{2} \)
13 \( 1 + 6.40T + 13T^{2} \)
17 \( 1 + 1.71T + 17T^{2} \)
19 \( 1 + 4.69T + 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 - 3.72iT - 29T^{2} \)
31 \( 1 + 2.42iT - 31T^{2} \)
37 \( 1 + 1.53T + 37T^{2} \)
41 \( 1 - 6.40T + 41T^{2} \)
43 \( 1 - 8.76iT - 43T^{2} \)
47 \( 1 - 2.42iT - 47T^{2} \)
53 \( 1 + 9.92T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 - 8.12T + 61T^{2} \)
67 \( 1 + 9.66T + 67T^{2} \)
71 \( 1 + 0.535T + 71T^{2} \)
73 \( 1 - 4.69T + 73T^{2} \)
79 \( 1 + 0.656iT - 79T^{2} \)
83 \( 1 - 9.38T + 83T^{2} \)
89 \( 1 - 5.74iT - 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−10.42559130726700926703278159673, −9.810386595214858039437812845792, −9.341124728405899250849334731738, −7.72482024125589977512383483934, −6.84610897963615364008111960735, −6.31576376053330826591924401664, −4.63877248072444868398292740141, −3.66692842127700928164494031563, −2.68237955737682788931278387416, −2.10740734230479865342104693711, 0.18618471182924529530121944750, 2.13371889931357589715649073937, 4.20026904683875892697907625092, 4.68553427220000582105945972279, 5.67683350566644022237585417821, 6.65512932680520490408225132738, 7.35146048517045910721042838830, 8.100101361309122378613194147879, 9.035295214234504969332520122517, 9.671181568281199128464968543524

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