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Properties

Label 2-847-11.3-c1-0-25
Degree $2$
Conductor $847$
Sign $0.266 - 0.963i$
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  + (0.809 − 0.587i)2-s + (0.618 + 1.90i)3-s + (−0.309 + 0.951i)4-s + (1.61 + 1.17i)5-s + (1.61 + 1.17i)6-s + (0.309 − 0.951i)7-s + (0.927 + 2.85i)8-s + (−0.809 + 0.587i)9-s + 2·10-s − 1.99·12-s + (3.23 − 2.35i)13-s + (−0.309 − 0.951i)14-s + (−1.23 + 3.80i)15-s + (0.809 + 0.587i)16-s + (3.23 + 2.35i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.356 + 1.09i)3-s + (−0.154 + 0.475i)4-s + (0.723 + 0.525i)5-s + (0.660 + 0.479i)6-s + (0.116 − 0.359i)7-s + (0.327 + 1.00i)8-s + (−0.269 + 0.195i)9-s + 0.632·10-s − 0.577·12-s + (0.897 − 0.652i)13-s + (−0.0825 − 0.254i)14-s + (−0.319 + 0.982i)15-s + (0.202 + 0.146i)16-s + (0.784 + 0.570i)17-s + (−0.0728 + 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.266 - 0.963i$
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: \((2,\ 847,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14110 + 1.62870i\)
\(L(\frac12)\) \(\approx\) \(2.14110 + 1.62870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.618 - 1.90i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-3.23 + 2.35i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.23 - 2.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.09 - 5.87i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.85 - 5.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.23 + 3.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (3.09 + 9.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.85 + 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.618 + 1.90i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + (-9.70 - 7.05i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.47 + 7.60i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.47 + 4.70i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-8.09 + 5.87i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.23494139218213315414604542842, −9.925304302037117316211401258071, −8.624832282101893197336165815049, −8.137245353097817380182442123901, −6.83468089114749104081318642101, −5.69196907920081428697993298372, −4.83336400590443012769423252647, −3.63521162200154162409984407459, −3.42572928598232855003846395413, −1.97682126617812537798111735642, 1.22171845552555633974236982378, 2.00648132285814175127456105490, 3.66357656518717343341756590537, 4.94510716776241660904599911135, 5.70370365490755499262595809528, 6.44529307839690765626180767451, 7.26307489520927824418314684200, 8.218523738520796703177157484021, 9.180304018161447928640428651392, 9.765194371661425904414712595854

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