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Properties

Label 2-847-7.4-c1-0-28
Degree $2$
Conductor $847$
Sign $-0.997 + 0.0687i$
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  + (−1.17 − 2.04i)2-s + (−1.32 + 2.29i)3-s + (−1.78 + 3.08i)4-s + (−1.66 − 2.88i)5-s + 6.25·6-s + (0.348 + 2.62i)7-s + 3.69·8-s + (−2.01 − 3.49i)9-s + (−3.92 + 6.79i)10-s + (−4.72 − 8.18i)12-s + 0.793·13-s + (4.94 − 3.80i)14-s + 8.82·15-s + (−0.789 − 1.36i)16-s + (0.338 − 0.585i)17-s + (−4.76 + 8.24i)18-s + ⋯
L(s)  = 1  + (−0.834 − 1.44i)2-s + (−0.765 + 1.32i)3-s + (−0.891 + 1.54i)4-s + (−0.743 − 1.28i)5-s + 2.55·6-s + (0.131 + 0.991i)7-s + 1.30·8-s + (−0.672 − 1.16i)9-s + (−1.24 + 2.14i)10-s + (−1.36 − 2.36i)12-s + 0.220·13-s + (1.32 − 1.01i)14-s + 2.27·15-s + (−0.197 − 0.342i)16-s + (0.0820 − 0.142i)17-s + (−1.12 + 1.94i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.997 + 0.0687i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (606, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.997 + 0.0687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00601914 - 0.174978i\)
\(L(\frac12)\) \(\approx\) \(0.00601914 - 0.174978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.348 - 2.62i)T \)
11 \( 1 \)
good2 \( 1 + (1.17 + 2.04i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.66 + 2.88i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.793T + 13T^{2} \)
17 \( 1 + (-0.338 + 0.585i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 6.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.254 - 0.440i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.93T + 29T^{2} \)
31 \( 1 + (-0.892 + 1.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.91 + 8.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.91T + 41T^{2} \)
43 \( 1 - 1.64T + 43T^{2} \)
47 \( 1 + (5.80 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.96 + 5.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.37 - 4.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.21 + 2.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.30 + 2.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + (0.256 - 0.443i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.16 + 8.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.10T + 83T^{2} \)
89 \( 1 + (0.872 + 1.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.03T + 97T^{2} \)
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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

−9.848049759340622647293375486034, −9.168144324487192330269428588468, −8.622398983241676261651384050780, −7.80523732367478121793407624826, −5.75561087286301804706370403429, −5.11446727065886671326614484142, −4.06744688598467135187788275716, −3.40001207526630988778446934654, −1.68901153262777491733068337605, −0.15191274596457372674601038298, 1.12831882387995428241761834266, 3.13796069453108360771216238231, 4.77848510663242249999978477422, 5.96141640181956468556488035107, 6.78642728991379763556908447335, 7.09064628418072535737738835875, 7.62543877152728585447097545929, 8.389020318317452635425513937304, 9.661430876487842119647521123865, 10.65652451768522136245460936054

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