↑

Properties

Label 2-847-11.5-c1-0-0
Degree $2$
Conductor $847$
Sign $-0.353 - 0.935i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.762 − 2.34i)2-s + (1.30 + 0.951i)3-s + (−3.30 + 2.40i)4-s + (−1.07 + 3.29i)5-s + (1.23 − 3.79i)6-s + (−0.809 + 0.587i)7-s + (4.16 + 3.02i)8-s + (−0.118 − 0.363i)9-s + 8.55·10-s − 6.60·12-s + (0.202 + 0.621i)13-s + (1.99 + 1.44i)14-s + (−4.53 + 3.29i)15-s + (1.39 − 4.29i)16-s + (0.351 − 1.08i)17-s + (−0.762 + 0.553i)18-s + ⋯
L(s)  = 1  + (−0.539 − 1.65i)2-s + (0.755 + 0.549i)3-s + (−1.65 + 1.20i)4-s + (−0.479 + 1.47i)5-s + (0.503 − 1.54i)6-s + (−0.305 + 0.222i)7-s + (1.47 + 1.06i)8-s + (−0.0393 − 0.121i)9-s + 2.70·10-s − 1.90·12-s + (0.0560 + 0.172i)13-s + (0.533 + 0.387i)14-s + (−1.17 + 0.851i)15-s + (0.348 − 1.07i)16-s + (0.0852 − 0.262i)17-s + (−0.179 + 0.130i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142246 + 0.205906i\)
\(L(\frac12)\) \(\approx\) \(0.142246 + 0.205906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.762 + 2.34i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.30 - 0.951i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.07 - 3.29i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.202 - 0.621i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.351 + 1.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.91 + 3.57i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + (3.70 - 2.68i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.864 - 2.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.355 + 0.258i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.77 + 3.46i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + (-0.489 - 0.355i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.03 - 9.34i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.37 - 0.995i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.11 - 6.52i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (1.67 - 5.15i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.42 + 3.93i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.820 + 2.52i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.06 - 6.36i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.698T + 89T^{2} \)
97 \( 1 + (4.59 + 14.1i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35517349859868778652504675618, −9.914512673975782227388686056303, −8.968273909944098518928047507847, −8.421785688822287070269361090202, −7.26758372494820425824498836058, −6.24256357376164607440571675119, −4.37353452325993314796017072670, −3.58217905365542351004902020104, −2.96006424249870870763704257958, −2.14378739484931639351575296631, 0.12787841212975035074173946483, 1.75571647358653125546503033566, 3.84018155340549182586821047004, 4.81484432644351324996646974981, 5.75364288222818759264326270909, 6.62678581781519177485372038981, 7.76212710362722782579690706802, 8.142303071943207614935684653886, 8.574259463357312045766793065956, 9.458701192186715702046194805847

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