Properties

Label 2-847-11.3-c1-0-2
Degree $2$
Conductor $847$
Sign $0.0694 + 0.997i$
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  + (−0.541 + 0.393i)2-s + (0.970 + 2.98i)3-s + (−0.479 + 1.47i)4-s + (−1.73 − 1.26i)5-s + (−1.70 − 1.23i)6-s + (0.309 − 0.951i)7-s + (−0.735 − 2.26i)8-s + (−5.54 + 4.03i)9-s + 1.43·10-s − 4.87·12-s + (−2.04 + 1.48i)13-s + (0.206 + 0.636i)14-s + (2.07 − 6.40i)15-s + (−1.22 − 0.887i)16-s + (1.44 + 1.05i)17-s + (1.41 − 4.36i)18-s + ⋯
L(s)  = 1  + (−0.383 + 0.278i)2-s + (0.560 + 1.72i)3-s + (−0.239 + 0.737i)4-s + (−0.775 − 0.563i)5-s + (−0.694 − 0.504i)6-s + (0.116 − 0.359i)7-s + (−0.259 − 0.799i)8-s + (−1.84 + 1.34i)9-s + 0.454·10-s − 1.40·12-s + (−0.567 + 0.412i)13-s + (0.0553 + 0.170i)14-s + (0.537 − 1.65i)15-s + (−0.305 − 0.221i)16-s + (0.351 + 0.255i)17-s + (0.334 − 1.02i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148844 - 0.138838i\)
\(L(\frac12)\) \(\approx\) \(0.148844 - 0.138838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.541 - 0.393i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.970 - 2.98i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.73 + 1.26i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.04 - 1.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.44 - 1.05i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.07 + 6.39i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 + (0.285 - 0.878i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.42 - 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.465 - 1.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.71 - 5.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.42T + 43T^{2} \)
47 \( 1 + (-1.35 - 4.18i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.539 - 0.392i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.113 - 0.350i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.05 + 2.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.902T + 67T^{2} \)
71 \( 1 + (-12.0 - 8.73i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.48 + 7.64i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.28 - 2.38i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.27 - 2.38i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.30T + 89T^{2} \)
97 \( 1 + (6.88 - 5.00i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.63659722766589382662754727492, −9.673897248478318722125112367800, −9.184609655493986202519501884983, −8.350548487490156820924460785605, −7.914472943596228023983716325260, −6.75507936070049744045145740126, −5.06744103758702404588998753186, −4.39587526470736431893912050387, −3.81378991743105917120843343578, −2.79294125532066416268524467339, 0.10278741092821085450821882856, 1.60397598623601393843750292142, 2.48024490705378551879592614920, 3.67222642603611401932235832217, 5.42357796330642981130844325042, 6.20285023605398355440440858270, 7.17345887364730330135615269423, 7.930685789108427226294229214617, 8.428267237704073557970804817328, 9.459208018017829064290374496004

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