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Properties

Label 2-847-11.4-c1-0-31
Degree $2$
Conductor $847$
Sign $0.898 - 0.437i$
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  + (1.12 + 0.814i)2-s + (−0.0378 + 0.116i)3-s + (−0.0244 − 0.0752i)4-s + (0.107 − 0.0781i)5-s + (−0.137 + 0.0997i)6-s + (0.309 + 0.951i)7-s + (0.890 − 2.74i)8-s + (2.41 + 1.75i)9-s + 0.184·10-s + 0.00969·12-s + (0.518 + 0.377i)13-s + (−0.428 + 1.31i)14-s + (0.00503 + 0.0154i)15-s + (3.10 − 2.25i)16-s + (1.15 − 0.837i)17-s + (1.27 + 3.93i)18-s + ⋯
L(s)  = 1  + (0.792 + 0.576i)2-s + (−0.0218 + 0.0672i)3-s + (−0.0122 − 0.0376i)4-s + (0.0481 − 0.0349i)5-s + (−0.0560 + 0.0407i)6-s + (0.116 + 0.359i)7-s + (0.314 − 0.968i)8-s + (0.804 + 0.584i)9-s + 0.0582·10-s + 0.00279·12-s + (0.143 + 0.104i)13-s + (−0.114 + 0.352i)14-s + (0.00129 + 0.00400i)15-s + (0.775 − 0.563i)16-s + (0.279 − 0.203i)17-s + (0.301 + 0.927i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.898 - 0.437i$
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: \((2,\ 847,\ (\ :1/2),\ 0.898 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48356 + 0.572815i\)
\(L(\frac12)\) \(\approx\) \(2.48356 + 0.572815i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 0.814i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.0378 - 0.116i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.107 + 0.0781i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.518 - 0.377i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.15 + 0.837i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.21 + 6.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.52 - 4.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.16 - 3.58i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.90 + 5.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 + (-2.93 + 9.02i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.539 - 0.391i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.52 - 7.78i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.68 - 5.58i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + (3.91 - 2.84i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.73 + 8.42i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.29 + 6.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.75 - 5.63i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (5.32 + 3.86i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.22361020677471261607015326897, −9.446205145012441787959119375548, −8.553958866509750924078827543800, −7.22131720782776689624102112983, −6.92579542615101081294531911309, −5.62587635130070722521716322914, −5.02310984152461676290472963368, −4.23648309130973617773546167580, −2.93910904999897071463127349048, −1.30431667498941607915438469787, 1.36153931294401205029095590918, 2.73954994682168980752203510835, 3.88366868578374534248576098866, 4.36691386262987549686828832496, 5.61676752246645897951308024605, 6.49649796122990120219346532466, 7.75700036241074079058372817351, 8.183942449952368458298471900817, 9.589001930433091797650453360310, 10.17480148410026314602538853090

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