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Properties

Label 2-847-77.76-c1-0-9
Degree $2$
Conductor $847$
Sign $0.143 + 0.989i$
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  + 2.20i·2-s + 1.86i·3-s − 2.87·4-s + 2.47i·5-s − 4.11·6-s + (−2.03 + 1.69i)7-s − 1.93i·8-s − 0.466·9-s − 5.46·10-s − 5.35i·12-s + 4.49·13-s + (−3.73 − 4.49i)14-s − 4.60·15-s − 1.48·16-s − 0.806·17-s − 1.03i·18-s + ⋯
L(s)  = 1  + 1.56i·2-s + 1.07i·3-s − 1.43·4-s + 1.10i·5-s − 1.67·6-s + (−0.769 + 0.638i)7-s − 0.683i·8-s − 0.155·9-s − 1.72·10-s − 1.54i·12-s + 1.24·13-s + (−0.997 − 1.20i)14-s − 1.18·15-s − 0.371·16-s − 0.195·17-s − 0.242i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.143 + 0.989i$
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: \((2,\ 847,\ (\ :1/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.959527 - 0.830794i\)
\(L(\frac12)\) \(\approx\) \(0.959527 - 0.830794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.03 - 1.69i)T \)
11 \( 1 \)
good2 \( 1 - 2.20iT - 2T^{2} \)
3 \( 1 - 1.86iT - 3T^{2} \)
5 \( 1 - 2.47iT - 5T^{2} \)
13 \( 1 - 4.49T + 13T^{2} \)
17 \( 1 + 0.806T + 17T^{2} \)
19 \( 1 + 6.84T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + 4.72iT - 29T^{2} \)
31 \( 1 - 7.02iT - 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 4.29T + 41T^{2} \)
43 \( 1 + 3.64iT - 43T^{2} \)
47 \( 1 - 3.90iT - 47T^{2} \)
53 \( 1 + 5.74T + 53T^{2} \)
59 \( 1 + 3.35iT - 59T^{2} \)
61 \( 1 - 6.69T + 61T^{2} \)
67 \( 1 + 3.99T + 67T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 - 8.11iT - 79T^{2} \)
83 \( 1 - 5.97T + 83T^{2} \)
89 \( 1 + 8.60iT - 89T^{2} \)
97 \( 1 + 3.00iT - 97T^{2} \)
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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

−10.81036995182224974154449127529, −9.759751918350987919856322842178, −8.990544670003972193235394640998, −8.384969642248024380771199893210, −7.13535086668535915307024836040, −6.46701309879316069745410513396, −5.92130333233953708139950418880, −4.78600323443954700021093726260, −3.84149649247657503716698351506, −2.74853539361955809153054203173, 0.68813362632081552857453603719, 1.38646856414852814107364666253, 2.62488812035927170879583411825, 3.88093463347385994106970581025, 4.59040202436372197253590404569, 6.13580584897988963065587727555, 6.89931699715318897737348009482, 8.094099399069340687829845999838, 8.939964607210083078418207895959, 9.553832776349910139871587617024

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