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Properties

Label 2-847-11.3-c1-0-3
Degree $2$
Conductor $847$
Sign $0.530 - 0.847i$
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  + (0.901 − 0.655i)2-s + (−0.883 − 2.71i)3-s + (−0.234 + 0.720i)4-s + (−2.79 − 2.03i)5-s + (−2.57 − 1.87i)6-s + (−0.309 + 0.951i)7-s + (0.949 + 2.92i)8-s + (−4.18 + 3.04i)9-s − 3.85·10-s + 2.16·12-s + (−1.66 + 1.21i)13-s + (0.344 + 1.05i)14-s + (−3.05 + 9.39i)15-s + (1.54 + 1.12i)16-s + (1.56 + 1.13i)17-s + (−1.78 + 5.48i)18-s + ⋯
L(s)  = 1  + (0.637 − 0.463i)2-s + (−0.510 − 1.56i)3-s + (−0.117 + 0.360i)4-s + (−1.25 − 0.908i)5-s + (−1.05 − 0.764i)6-s + (−0.116 + 0.359i)7-s + (0.335 + 1.03i)8-s + (−1.39 + 1.01i)9-s − 1.21·10-s + 0.625·12-s + (−0.462 + 0.335i)13-s + (0.0920 + 0.283i)14-s + (−0.788 + 2.42i)15-s + (0.386 + 0.280i)16-s + (0.379 + 0.275i)17-s + (−0.419 + 1.29i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.530 - 0.847i$
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: \((2,\ 847,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245528 + 0.136053i\)
\(L(\frac12)\) \(\approx\) \(0.245528 + 0.136053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.901 + 0.655i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.883 + 2.71i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.79 + 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.66 - 1.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.56 - 1.13i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.501 + 1.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.807T + 23T^{2} \)
29 \( 1 + (2.46 - 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.637 - 0.463i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.10 + 9.56i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.657 - 2.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 + (-2.33 - 7.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.75 - 6.36i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.01 - 3.13i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.871 + 0.632i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.40T + 67T^{2} \)
71 \( 1 + (-2.57 - 1.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.378 - 1.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.67 - 5.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.0 + 9.44i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.43T + 89T^{2} \)
97 \( 1 + (5.23 - 3.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.98016421929504755003229425636, −9.165768909528476810854488368748, −8.381106161040050364558754820562, −7.68393245222323846485485980555, −7.12075404046962908696752339266, −5.82764165516003163593394437361, −4.94438351875170211897448418955, −4.00143604417472118645869004026, −2.76768203210705341769411003140, −1.47141429232165499922229230985, 0.12233385855268265707194014132, 3.18782459427980286191500938577, 3.93776066358680995056288399551, 4.56227781449835369129423017309, 5.46337357396655375847405880877, 6.38691242294073764090520745756, 7.30377500302318816407125175878, 8.249643464820085651542649931383, 9.790441077297322332115915814840, 9.935061177413357893086871536454

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