↑

Properties

Label 2-847-847.20-c0-0-0
Degree $2$
Conductor $847$
Sign $-0.802 + 0.596i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.982 − 1.43i)2-s + (−0.737 − 1.89i)4-s + (−0.870 + 0.491i)7-s + (−1.75 − 0.407i)8-s + (0.309 − 0.951i)9-s + (−0.0285 − 0.999i)11-s + (−0.149 + 1.73i)14-s + (−0.810 + 0.743i)16-s + (−1.06 − 1.37i)18-s + (−1.46 − 0.941i)22-s + (0.164 + 0.361i)23-s + (0.696 + 0.717i)25-s + (1.57 + 1.28i)28-s + (−1.28 + 1.32i)29-s + (0.0165 + 0.114i)32-s + ⋯
L(s)  = 1  + (0.982 − 1.43i)2-s + (−0.737 − 1.89i)4-s + (−0.870 + 0.491i)7-s + (−1.75 − 0.407i)8-s + (0.309 − 0.951i)9-s + (−0.0285 − 0.999i)11-s + (−0.149 + 1.73i)14-s + (−0.810 + 0.743i)16-s + (−1.06 − 1.37i)18-s + (−1.46 − 0.941i)22-s + (0.164 + 0.361i)23-s + (0.696 + 0.717i)25-s + (1.57 + 1.28i)28-s + (−1.28 + 1.32i)29-s + (0.0165 + 0.114i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.802 + 0.596i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ -0.802 + 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432665632\)
\(L(\frac12)\) \(\approx\) \(1.432665632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.870 - 0.491i)T \)
11 \( 1 + (0.0285 + 0.999i)T \)
good2 \( 1 + (-0.982 + 1.43i)T + (-0.362 - 0.931i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.696 - 0.717i)T^{2} \)
13 \( 1 + (-0.198 + 0.980i)T^{2} \)
17 \( 1 + (-0.610 + 0.791i)T^{2} \)
19 \( 1 + (0.564 - 0.825i)T^{2} \)
23 \( 1 + (-0.164 - 0.361i)T + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (1.28 - 1.32i)T + (-0.0285 - 0.999i)T^{2} \)
31 \( 1 + (0.870 - 0.491i)T^{2} \)
37 \( 1 + (-1.61 - 0.0923i)T + (0.993 + 0.113i)T^{2} \)
41 \( 1 + (-0.774 + 0.633i)T^{2} \)
43 \( 1 + (-0.964 - 1.11i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (0.254 + 0.967i)T^{2} \)
53 \( 1 + (-1.45 - 1.33i)T + (0.0855 + 0.996i)T^{2} \)
59 \( 1 + (-0.774 - 0.633i)T^{2} \)
61 \( 1 + (0.362 - 0.931i)T^{2} \)
67 \( 1 + (1.72 + 0.505i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (1.52 - 0.264i)T + (0.941 - 0.336i)T^{2} \)
73 \( 1 + (0.921 + 0.389i)T^{2} \)
79 \( 1 + (0.262 + 0.435i)T + (-0.466 + 0.884i)T^{2} \)
83 \( 1 + (-0.974 - 0.226i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.696 + 0.717i)T^{2} \)
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

−10.30459865866445206133715737204, −9.305088416410822483612738392224, −9.007362795241501152662178169691, −7.32785573150195462187704267106, −6.10751241585323476580591008466, −5.58936477903111138642427677563, −4.31608114869533770022757566661, −3.37491046093295276774582140106, −2.82998261514683731063331946513, −1.21213632709205453724479205045, 2.51470239469701233894490941438, 3.98547378000708078007707677072, 4.52331905133506873265343078818, 5.56614094775051278106345542975, 6.40492122356057497798821487828, 7.29837316190874188137417462795, 7.64363173987245453311272084515, 8.767961714921406949761257182311, 9.840778136462553399913581449573, 10.58466222627475860535678961229

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