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Properties

Label 2-847-11.9-c1-0-50
Degree $2$
Conductor $847$
Sign $-0.780 - 0.625i$
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  + (0.690 − 2.12i)2-s + (1 − 0.726i)3-s + (−2.42 − 1.76i)4-s + (−0.618 − 1.90i)5-s + (−0.854 − 2.62i)6-s + (−0.809 − 0.587i)7-s + (−1.80 + 1.31i)8-s + (−0.454 + 1.40i)9-s − 4.47·10-s − 3.70·12-s + (1 − 3.07i)13-s + (−1.80 + 1.31i)14-s + (−2 − 1.45i)15-s + (−0.309 − 0.951i)16-s + (−1 − 3.07i)17-s + (2.66 + 1.93i)18-s + ⋯
L(s)  = 1  + (0.488 − 1.50i)2-s + (0.577 − 0.419i)3-s + (−1.21 − 0.881i)4-s + (−0.276 − 0.850i)5-s + (−0.348 − 1.07i)6-s + (−0.305 − 0.222i)7-s + (−0.639 + 0.464i)8-s + (−0.151 + 0.466i)9-s − 1.41·10-s − 1.07·12-s + (0.277 − 0.853i)13-s + (−0.483 + 0.351i)14-s + (−0.516 − 0.375i)15-s + (−0.0772 − 0.237i)16-s + (−0.242 − 0.746i)17-s + (0.627 + 0.456i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613565 + 1.74599i\)
\(L(\frac12)\) \(\approx\) \(0.613565 + 1.74599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.690 + 2.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1 + 0.726i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.618 + 1.90i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1 + 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1 + 3.07i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.23 - 3.80i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (6.85 + 4.97i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.854 - 2.62i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.85 - 4.97i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.09 + 6.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (2.23 - 1.62i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.145 - 0.449i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1 - 0.726i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 + 6.88i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + (3.23 + 9.95i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.618 - 0.449i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.76 - 8.50i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.52 + 10.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (-5.38 + 16.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.802181822961634367667861639312, −9.004611732568921726420086192029, −8.163187620499137499789279236036, −7.36015275805447262241335499466, −5.85673746316624796161782468964, −4.80788535520674662589961419632, −3.99074006204298272687537273804, −2.94253833433674052783226131979, −2.02506923175384044865016280440, −0.71719218579291389814672361364, 2.53450686856323812736009310196, 3.79284066685014725315002602463, 4.35279908299603322761005404691, 5.73161767435830328679426959910, 6.49666165944835380170292703926, 7.04441125836766200090731439136, 8.010087530758647901081656797897, 8.956399335691149967495942883833, 9.333638947151204885432340365972, 10.80132176686793335575758951110

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