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Properties

Label 2-847-11.4-c1-0-6
Degree $2$
Conductor $847$
Sign $-0.859 - 0.511i$
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  + (−0.927 + 2.85i)3-s + (−0.618 − 1.90i)4-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + (−4.85 − 3.52i)9-s + 6·12-s + (3.23 + 2.35i)13-s + (0.927 + 2.85i)15-s + (−3.23 + 2.35i)16-s + (−1.61 + 1.17i)17-s + (−1.85 + 5.70i)19-s + (−1.61 − 1.17i)20-s + 3·21-s − 5·23-s + (−1.23 + 3.80i)25-s + ⋯
L(s)  = 1  + (−0.535 + 1.64i)3-s + (−0.309 − 0.951i)4-s + (0.361 − 0.262i)5-s + (−0.116 − 0.359i)7-s + (−1.61 − 1.17i)9-s + 1.73·12-s + (0.897 + 0.652i)13-s + (0.239 + 0.736i)15-s + (−0.809 + 0.587i)16-s + (−0.392 + 0.285i)17-s + (−0.425 + 1.30i)19-s + (−0.361 − 0.262i)20-s + 0.654·21-s − 1.04·23-s + (−0.247 + 0.760i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.859 - 0.511i$
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: \((2,\ 847,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185883 + 0.675850i\)
\(L(\frac12)\) \(\approx\) \(0.185883 + 0.675850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.927 - 2.85i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-3.23 - 2.35i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.61 - 1.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.85 - 5.70i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 + (-3.09 - 9.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.54 + 4.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.618 - 1.90i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-2.47 + 7.60i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.85 - 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.927 - 2.85i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.61 + 1.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (0.809 - 0.587i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.09 - 9.51i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.85 + 3.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.70 - 7.05i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-4.04 - 2.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.35390066447151499866308717595, −9.953118478650024804061516136079, −9.057197029011564538102728639270, −8.524732385927276877734063511625, −6.71843770074205831825508880950, −5.83414094097184979341264991970, −5.29667401117005760025988128442, −4.24545614410316390284225995878, −3.69280650178484932255531239105, −1.59698148138712819478518166525, 0.36429114385404090000695432094, 2.11043495212053107590904379756, 2.95038797261019570639302863580, 4.47588821917921845466291307550, 5.80029400882931160556507219633, 6.45746843615157233125644712578, 7.18583560045705690708390309795, 8.202350376194745578808427725414, 8.508697939236556891074884935004, 9.804263758573919835765140791742

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