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Properties

Label 2-847-11.4-c1-0-7
Degree $2$
Conductor $847$
Sign $0.0694 - 0.997i$
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  + (−2.06 − 1.49i)2-s + (−0.126 + 0.390i)3-s + (1.39 + 4.29i)4-s + (−2.77 + 2.01i)5-s + (0.847 − 0.615i)6-s + (0.309 + 0.951i)7-s + (1.98 − 6.09i)8-s + (2.29 + 1.66i)9-s + 8.75·10-s − 1.85·12-s + (1.75 + 1.27i)13-s + (0.788 − 2.42i)14-s + (−0.435 − 1.33i)15-s + (−5.93 + 4.31i)16-s + (3.65 − 2.65i)17-s + (−2.23 − 6.87i)18-s + ⋯
L(s)  = 1  + (−1.45 − 1.06i)2-s + (−0.0732 + 0.225i)3-s + (0.697 + 2.14i)4-s + (−1.24 + 0.901i)5-s + (0.346 − 0.251i)6-s + (0.116 + 0.359i)7-s + (0.700 − 2.15i)8-s + (0.763 + 0.554i)9-s + 2.76·10-s − 0.534·12-s + (0.488 + 0.354i)13-s + (0.210 − 0.648i)14-s + (−0.112 − 0.345i)15-s + (−1.48 + 1.07i)16-s + (0.886 − 0.643i)17-s + (−0.526 − 1.61i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0694 - 0.997i$
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: \((2,\ 847,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351238 + 0.327627i\)
\(L(\frac12)\) \(\approx\) \(0.351238 + 0.327627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 + 1.49i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.126 - 0.390i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.77 - 2.01i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.75 - 1.27i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.65 + 2.65i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.748 - 2.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.648T + 23T^{2} \)
29 \( 1 + (0.387 + 1.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.50 - 4.72i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.54 - 4.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.810 - 2.49i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + (1.55 - 4.79i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.8 + 7.87i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.37 - 7.29i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.5 - 8.42i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + (3.42 - 2.48i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.83 + 5.63i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.89 + 5.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.77 + 4.92i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + (-7.04 - 5.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.36700021884947006715439326320, −9.815443532930283474770357109462, −8.770004430103096628361754868586, −7.910749388406510791631626258087, −7.51623412303531736450904956126, −6.49603067712811588583269304858, −4.65842991431679835759164833128, −3.56075272055757801589198958422, −2.80273192635350309340538150264, −1.38325038110593202742434697265, 0.46617443952981662768579523381, 1.35934947020327228842754942144, 3.73634606741282391267902705013, 4.79801351102475817701713324523, 5.98977991008072066602908021194, 6.85549185327360712622937651294, 7.76313673631800859102527164327, 8.060431852669994963004583240131, 8.938758383760376530299378677952, 9.696254814353523908238896694280

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