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Properties

Label 2-847-77.76-c1-0-21
Degree $2$
Conductor $847$
Sign $0.479 + 0.877i$
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  − 2.06i·2-s + 1.78i·3-s − 2.25·4-s + 0.290i·5-s + 3.68·6-s + (−2.64 − 0.129i)7-s + 0.517i·8-s − 0.200·9-s + 0.598·10-s − 4.02i·12-s + 2.85·13-s + (−0.267 + 5.44i)14-s − 0.519·15-s − 3.43·16-s + 6.77·17-s + 0.413i·18-s + ⋯
L(s)  = 1  − 1.45i·2-s + 1.03i·3-s − 1.12·4-s + 0.129i·5-s + 1.50·6-s + (−0.998 − 0.0491i)7-s + 0.183i·8-s − 0.0668·9-s + 0.189·10-s − 1.16i·12-s + 0.792·13-s + (−0.0716 + 1.45i)14-s − 0.134·15-s − 0.858·16-s + 1.64·17-s + 0.0975i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29497 - 0.767879i\)
\(L(\frac12)\) \(\approx\) \(1.29497 - 0.767879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.64 + 0.129i)T \)
11 \( 1 \)
good2 \( 1 + 2.06iT - 2T^{2} \)
3 \( 1 - 1.78iT - 3T^{2} \)
5 \( 1 - 0.290iT - 5T^{2} \)
13 \( 1 - 2.85T + 13T^{2} \)
17 \( 1 - 6.77T + 17T^{2} \)
19 \( 1 - 2.49T + 19T^{2} \)
23 \( 1 + 4.08T + 23T^{2} \)
29 \( 1 + 6.91iT - 29T^{2} \)
31 \( 1 - 5.12iT - 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 + 2.51iT - 43T^{2} \)
47 \( 1 + 7.65iT - 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 4.12iT - 59T^{2} \)
61 \( 1 + 6.06T + 61T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 13.9iT - 79T^{2} \)
83 \( 1 + 0.375T + 83T^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 - 5.85iT - 97T^{2} \)
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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

−10.24580962134402509540219824081, −9.580943613480681521089866061254, −8.956562029849482552978078942327, −7.63178786382241095624032932195, −6.42379794500607406673884162860, −5.35459610644863325554295447871, −4.07854276914334821107272345871, −3.57560051044577821570329798831, −2.70192276342347330097255972981, −1.02192165400403266229875592530, 1.08027496949742517941484825123, 2.84836888397326366782856453096, 4.21247440116989854373110349905, 5.68305829130942582233738465884, 6.04009162187997170258788336523, 6.98721743072394430627226524430, 7.54599503917670026437723830461, 8.257343571311072513451978453260, 9.217257925123920844884825374607, 10.02623739494785471148286489431

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