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Properties

Label 2-847-11.3-c1-0-5
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  + (−0.927 − 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

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

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

\(L(1)\) \(\approx\) \(0.525547 + 0.399775i\)
\(L(\frac12)\) \(\approx\) \(0.525547 + 0.399775i\)
\(L(\frac{3}{2})\) 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

−10.48024670499657144807090778239, −9.450455904672403981530183413665, −8.272216169503526229222219904402, −7.69368580106330520897266611079, −7.06043921021673310240955918594, −6.25218295341912499298566403655, −5.26844750815053186609389963032, −3.93412662973625388084950185321, −2.60062299094316533223165483521, −1.52456574736774742432399877651, 0.34350281464779691706024284713, 2.43726768402243278348847815986, 3.94173969165015921204476229044, 4.80913474218338749643048125429, 5.52314628791857521767730749047, 5.90845286118685222436989965085, 7.45528048174537683151877370280, 8.860729206698194495605240030984, 9.470912387488351533203650978441, 9.906498639828204344040687960287

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