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Properties

Label 2-847-847.111-c0-0-0
Degree $2$
Conductor $847$
Sign $0.218 + 0.975i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 0.708i)2-s + (0.297 + 0.650i)4-s + (−0.654 − 0.755i)7-s + (−0.0530 + 0.368i)8-s + 9-s + (0.841 + 0.540i)11-s + (0.186 + 1.29i)14-s + (0.788 − 0.909i)16-s + (−1.10 − 0.708i)18-s + (−0.544 − 1.19i)22-s + (0.857 − 0.989i)23-s + (−0.959 + 0.281i)25-s + (0.297 − 0.650i)28-s + (0.273 + 0.0801i)29-s + (−1.15 + 0.339i)32-s + ⋯
L(s)  = 1  + (−1.10 − 0.708i)2-s + (0.297 + 0.650i)4-s + (−0.654 − 0.755i)7-s + (−0.0530 + 0.368i)8-s + 9-s + (0.841 + 0.540i)11-s + (0.186 + 1.29i)14-s + (0.788 − 0.909i)16-s + (−1.10 − 0.708i)18-s + (−0.544 − 1.19i)22-s + (0.857 − 0.989i)23-s + (−0.959 + 0.281i)25-s + (0.297 − 0.650i)28-s + (0.273 + 0.0801i)29-s + (−1.15 + 0.339i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.218 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5730909936\)
\(L(\frac12)\) \(\approx\) \(0.5730909936\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.654 + 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
good2 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
23 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (-0.830 + 1.81i)T + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (-0.415 + 0.909i)T^{2} \)
53 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−10.40356109255954480982475027726, −9.312949011037884713756574738323, −9.037903907575976546022637877613, −7.62515432012125337677207230350, −7.14645263397982021229165516552, −6.05200256040222184025519306215, −4.58448276893958204128327603803, −3.68154530721578523172607219704, −2.25587779715616670595027667891, −1.02340311962395258080551571696, 1.35661443771431512359338153740, 3.15072789466233822118957036669, 4.25355150527001718713415106502, 5.73160088048934359847143493212, 6.51201390519640281070925776336, 7.17415064709445722462457166192, 8.155449682431465251361666282438, 8.869626765158684164480847110947, 9.708344294141323305841223925754, 9.973964660613159295314612336344

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