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Properties

Label 2-847-11.9-c1-0-17
Degree $2$
Conductor $847$
Sign $-0.569 - 0.821i$
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  + (−0.453 + 1.39i)2-s + (1.30 − 0.951i)3-s + (−0.122 − 0.0889i)4-s + (0.144 + 0.443i)5-s + (0.733 + 2.25i)6-s + (0.809 + 0.587i)7-s + (−2.19 + 1.59i)8-s + (−0.118 + 0.363i)9-s − 0.684·10-s − 0.244·12-s + (−0.488 + 1.50i)13-s + (−1.18 + 0.862i)14-s + (0.610 + 0.443i)15-s + (−1.32 − 4.07i)16-s + (1.61 + 4.97i)17-s + (−0.453 − 0.329i)18-s + ⋯
L(s)  = 1  + (−0.320 + 0.986i)2-s + (0.755 − 0.549i)3-s + (−0.0612 − 0.0444i)4-s + (0.0645 + 0.198i)5-s + (0.299 + 0.921i)6-s + (0.305 + 0.222i)7-s + (−0.775 + 0.563i)8-s + (−0.0393 + 0.121i)9-s − 0.216·10-s − 0.0706·12-s + (−0.135 + 0.417i)13-s + (−0.317 + 0.230i)14-s + (0.157 + 0.114i)15-s + (−0.330 − 1.01i)16-s + (0.391 + 1.20i)17-s + (−0.106 − 0.0776i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.569 - 0.821i$
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: \((2,\ 847,\ (\ :1/2),\ -0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773552 + 1.47779i\)
\(L(\frac12)\) \(\approx\) \(0.773552 + 1.47779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.453 - 1.39i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.144 - 0.443i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.488 - 1.50i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.61 - 4.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.41 - 2.48i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 + (2.20 + 1.59i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.399 + 1.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.57 - 1.14i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.842 + 0.611i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-5.17 + 3.75i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.08 - 12.5i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.96 - 5.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.70 + 14.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + (-3.01 - 9.26i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.7 - 7.82i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.10 + 3.40i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.33 + 16.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 + (0.835 - 2.57i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.40847100144516929402482474378, −9.202059914806710591795915666599, −8.460519677390421636357273244823, −7.966077610203784490195584116811, −7.24582257322145240689493514549, −6.31943551287548623634499885172, −5.60807487593410831066860763985, −4.18758916412974002628084998388, −2.79918026244723281152199411402, −1.89144300016251709289998031199, 0.816100469152108448182357842755, 2.34927234535923147752009460123, 3.14685542323734084882103498293, 4.12203939360168752339747225631, 5.26175664071336200264556454543, 6.48155911866332443633756836920, 7.52368813983917566927084504137, 8.631923821407954822498398159206, 9.258269953673898525956909917420, 9.837099718413747451187523920104

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