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Properties

Label 2-847-11.4-c1-0-5
Degree $2$
Conductor $847$
Sign $0.751 - 0.659i$
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  + (−2.11 − 1.53i)2-s + (0.190 − 0.587i)3-s + (1.5 + 4.61i)4-s + (−0.809 + 0.587i)5-s + (−1.30 + 0.951i)6-s + (−0.309 − 0.951i)7-s + (2.30 − 7.10i)8-s + (2.11 + 1.53i)9-s + 2.61·10-s + 3·12-s + (2.61 + 1.90i)13-s + (−0.809 + 2.48i)14-s + (0.190 + 0.587i)15-s + (−7.97 + 5.79i)16-s + (−6.54 + 4.75i)17-s + (−2.11 − 6.51i)18-s + ⋯
L(s)  = 1  + (−1.49 − 1.08i)2-s + (0.110 − 0.339i)3-s + (0.750 + 2.30i)4-s + (−0.361 + 0.262i)5-s + (−0.534 + 0.388i)6-s + (−0.116 − 0.359i)7-s + (0.816 − 2.51i)8-s + (0.706 + 0.512i)9-s + 0.827·10-s + 0.866·12-s + (0.726 + 0.527i)13-s + (−0.216 + 0.665i)14-s + (0.0493 + 0.151i)15-s + (−1.99 + 1.44i)16-s + (−1.58 + 1.15i)17-s + (−0.499 − 1.53i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407156 + 0.153217i\)
\(L(\frac12)\) \(\approx\) \(0.407156 + 0.153217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 + 1.53i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-2.61 - 1.90i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.54 - 4.75i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.92 + 5.93i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 + (-0.736 - 2.26i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.190 + 0.138i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.763 + 2.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.45 - 10.6i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + (1.35 - 4.16i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.73 - 2.71i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0278 + 0.0857i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.35 - 3.16i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 + (-3.97 + 2.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.01 - 9.28i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.97 - 5.06i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.66 - 6.29i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.145T + 89T^{2} \)
97 \( 1 + (5.66 + 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.29223042083393320398177663801, −9.538664703612272516624328896455, −8.644789005544705358220567361093, −8.047930720519814554224072083058, −7.13395298986681956469320492043, −6.53815067934900777325441222114, −4.45009931969146871252093893625, −3.56265589582729985794042488905, −2.28956625642690583184774569728, −1.37716706471016319023995759515, 0.36057832018434196405194860148, 1.91194235687352859935992434511, 3.79367571489340229552461354673, 5.05078249331395879736633917242, 6.11291769978806421755165106629, 6.76990851634504468479773042617, 7.74253810169022737785626945664, 8.460526954584575604307388428832, 9.059128755992803956544528664835, 9.952661979140163112375758296353

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