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Properties

Label 2-847-11.3-c1-0-44
Degree $2$
Conductor $847$
Sign $-0.266 + 0.963i$
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  + (1.80 − 1.31i)2-s + (−0.381 − 1.17i)3-s + (0.927 − 2.85i)4-s + (1.61 + 1.17i)5-s + (−2.23 − 1.62i)6-s + (−0.309 + 0.951i)7-s + (−0.690 − 2.12i)8-s + (1.19 − 0.865i)9-s + 4.47·10-s − 3.70·12-s + (2.61 − 1.90i)13-s + (0.690 + 2.12i)14-s + (0.763 − 2.35i)15-s + (0.809 + 0.587i)16-s + (−2.61 − 1.90i)17-s + (1.01 − 3.13i)18-s + ⋯
L(s)  = 1  + (1.27 − 0.929i)2-s + (−0.220 − 0.678i)3-s + (0.463 − 1.42i)4-s + (0.723 + 0.525i)5-s + (−0.912 − 0.663i)6-s + (−0.116 + 0.359i)7-s + (−0.244 − 0.751i)8-s + (0.396 − 0.288i)9-s + 1.41·10-s − 1.07·12-s + (0.726 − 0.527i)13-s + (0.184 + 0.568i)14-s + (0.197 − 0.607i)15-s + (0.202 + 0.146i)16-s + (−0.634 − 0.461i)17-s + (0.239 − 0.737i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02275 - 2.65912i\)
\(L(\frac12)\) \(\approx\) \(2.02275 - 2.65912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.31i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.381 + 1.17i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-2.61 + 1.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.61 + 1.90i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2 + 6.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (2.61 - 8.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.23 + 1.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.61 - 8.05i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.47 - 10.6i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-0.854 - 2.62i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.381 + 0.277i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.381 - 1.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.85 + 4.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + (-8.47 - 6.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.236 + 0.726i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.23 - 5.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.23 + 6.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (14.0 - 10.2i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.25844282850596857842432047084, −9.386570291074995248533504206537, −8.259653363155742317728322939220, −6.77086216135483282032309102345, −6.44588502510846930191913397605, −5.39836702441688353034468488193, −4.52799714307787666467962617319, −3.23153222755118885349531882657, −2.46638243204561322734470763753, −1.32000787275111845634871855860, 1.85196264082905807521776008050, 3.80431242425887039774423735250, 4.15823573132153968983829558378, 5.23840742409575886587554999579, 5.84091806396628151621408007185, 6.66740717151559082144934949971, 7.62143053711930633866854070943, 8.660005644919577299855900132380, 9.660238240262594142939935938114, 10.39790157205000488282162834358

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