Properties

Label 2-847-11.9-c1-0-1
Degree $2$
Conductor $847$
Sign $-0.605 + 0.795i$
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  + (−0.402 + 1.23i)2-s + (−1.05 + 0.765i)3-s + (0.244 + 0.177i)4-s + (−1.11 − 3.42i)5-s + (−0.524 − 1.61i)6-s + (0.809 + 0.587i)7-s + (−2.42 + 1.76i)8-s + (−0.402 + 1.23i)9-s + 4.69·10-s − 0.394·12-s + (−0.187 + 0.575i)13-s + (−1.05 + 0.765i)14-s + (3.80 + 2.76i)15-s + (−1.02 − 3.14i)16-s + (1.94 + 5.99i)17-s + (−1.37 − 0.997i)18-s + ⋯
L(s)  = 1  + (−0.284 + 0.876i)2-s + (−0.608 + 0.442i)3-s + (0.122 + 0.0889i)4-s + (−0.498 − 1.53i)5-s + (−0.214 − 0.658i)6-s + (0.305 + 0.222i)7-s + (−0.858 + 0.623i)8-s + (−0.134 + 0.413i)9-s + 1.48·10-s − 0.113·12-s + (−0.0518 + 0.159i)13-s + (−0.281 + 0.204i)14-s + (0.981 + 0.712i)15-s + (−0.255 − 0.785i)16-s + (0.472 + 1.45i)17-s + (−0.323 − 0.235i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.605 + 0.795i$
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: \((2,\ 847,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0954400 - 0.192514i\)
\(L(\frac12)\) \(\approx\) \(0.0954400 - 0.192514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.402 - 1.23i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.05 - 0.765i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.11 + 3.42i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.187 - 0.575i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.94 - 5.99i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.42 + 1.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 + (6.71 + 4.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.45 + 5.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.66 - 4.11i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.30T + 43T^{2} \)
47 \( 1 + (7.20 - 5.23i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.646 - 1.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.33 + 6.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.215 + 0.663i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.51T + 67T^{2} \)
71 \( 1 + (0.215 + 0.663i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.04 - 2.93i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.45 - 4.46i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.927 - 2.85i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + (-1.11 + 3.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.80507962240783351088668585212, −9.649139314252215677628286910486, −8.830013844823603357831340903031, −7.995895791002138686567369463834, −7.73391687316774827326648603883, −6.17568012573830696070619238070, −5.56396556210601190251746999365, −4.78843855750364345036820536173, −3.78585468440581452774022362349, −1.88230470545999933646424423264, 0.11804911461484907480070043421, 1.69826877095568580312838663160, 3.05297974978714713277355616188, 3.59789004871286936346042687299, 5.37582841154722148216846525924, 6.33223326813622793734828192988, 7.06129588733101442911483812466, 7.64158468417295235110162228840, 9.088596468987496533303374984587, 10.05401390980560964118400827216

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