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Properties

Label 2-847-77.41-c1-0-10
Degree $2$
Conductor $847$
Sign $0.486 - 0.873i$
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  + (−0.513 − 0.707i)2-s + (1.26 − 0.410i)3-s + (0.381 − 1.17i)4-s + (−0.780 + 1.07i)5-s + (−0.939 − 0.682i)6-s + (0.169 + 2.64i)7-s + (−2.68 + 0.874i)8-s + (−0.999 + 0.726i)9-s + 1.16·10-s − 1.64i·12-s + (−4.91 + 3.57i)13-s + (1.77 − 1.47i)14-s + (−0.545 + 1.67i)15-s + (3.39 + 2.46i)17-s + (1.02 + 0.333i)18-s + (0.939 + 2.89i)19-s + ⋯
L(s)  = 1  + (−0.363 − 0.499i)2-s + (0.729 − 0.236i)3-s + (0.190 − 0.587i)4-s + (−0.349 + 0.480i)5-s + (−0.383 − 0.278i)6-s + (0.0641 + 0.997i)7-s + (−0.951 + 0.309i)8-s + (−0.333 + 0.242i)9-s + 0.367·10-s − 0.473i·12-s + (−1.36 + 0.990i)13-s + (0.475 − 0.394i)14-s + (−0.140 + 0.433i)15-s + (0.824 + 0.598i)17-s + (0.242 + 0.0786i)18-s + (0.215 + 0.663i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.486 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875792 + 0.514759i\)
\(L(\frac12)\) \(\approx\) \(0.875792 + 0.514759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.169 - 2.64i)T \)
11 \( 1 \)
good2 \( 1 + (0.513 + 0.707i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.26 + 0.410i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.780 - 1.07i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.91 - 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.39 - 2.46i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.939 - 2.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (6.52 + 2.12i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.30 + 4.55i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.927 - 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.939 - 2.89i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.61iT - 43T^{2} \)
47 \( 1 + (-9.14 + 2.96i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.61 + 1.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-6.91 - 2.24i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.45 + 1.78i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (0.809 + 0.587i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.09 + 6.46i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.49 + 3.43i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.01 - 5.09i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 18.5iT - 89T^{2} \)
97 \( 1 + (3.30 + 4.55i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.26266516128659821286480661310, −9.355388300784104124375093464836, −8.947335053732611610085837482961, −7.83994441121394257397250228858, −7.12182601545655379610820095276, −5.88141200247483274681369236724, −5.19143101879679311423966040117, −3.54293522412170886558630191752, −2.51447012031473359026210129710, −1.84491658823773587090554604131, 0.47871080155450046536724906883, 2.76050850656098339510556910759, 3.44264919046498371160063371197, 4.56603451308110392679159262723, 5.66484462250562786619543897462, 7.21527046880684390189705988813, 7.40700946729502369034527879628, 8.292017760839522215411067180761, 9.079841081274649638821079123066, 9.726978156213341580440971175921

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