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Properties

Label 2-847-11.5-c1-0-21
Degree $2$
Conductor $847$
Sign $-0.923 - 0.382i$
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  + (0.394 + 1.21i)2-s + (2.08 + 1.51i)3-s + (0.299 − 0.217i)4-s + (−1.26 + 3.89i)5-s + (−1.01 + 3.12i)6-s + (−0.809 + 0.587i)7-s + (2.44 + 1.77i)8-s + (1.12 + 3.45i)9-s − 5.22·10-s + 0.954·12-s + (−1.35 − 4.17i)13-s + (−1.03 − 0.750i)14-s + (−8.52 + 6.19i)15-s + (−0.964 + 2.96i)16-s + (1.29 − 3.98i)17-s + (−3.75 + 2.72i)18-s + ⋯
L(s)  = 1  + (0.278 + 0.858i)2-s + (1.20 + 0.874i)3-s + (0.149 − 0.108i)4-s + (−0.565 + 1.74i)5-s + (−0.414 + 1.27i)6-s + (−0.305 + 0.222i)7-s + (0.865 + 0.628i)8-s + (0.374 + 1.15i)9-s − 1.65·10-s + 0.275·12-s + (−0.376 − 1.15i)13-s + (−0.276 − 0.200i)14-s + (−2.20 + 1.59i)15-s + (−0.241 + 0.742i)16-s + (0.314 − 0.966i)17-s + (−0.885 + 0.643i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 847,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529263 + 2.66133i\)
\(L(\frac12)\) \(\approx\) \(0.529263 + 2.66133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.394 - 1.21i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.08 - 1.51i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.26 - 3.89i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.35 + 4.17i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.29 + 3.98i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.01 - 0.734i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + (-1.56 + 1.13i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.482 + 1.48i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.579 + 0.421i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.88 + 2.82i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.35T + 43T^{2} \)
47 \( 1 + (-8.46 - 6.15i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.22 + 3.77i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.1 - 8.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.62 - 11.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 + (-0.0674 + 0.207i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.87 - 6.44i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.40 + 4.33i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.758 + 2.33i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.20T + 89T^{2} \)
97 \( 1 + (3.37 + 10.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.41300741204919031018141232508, −9.838391708242779756397150065172, −8.745050060361078722387278728840, −7.59092387526963411652709380650, −7.44366609416064680594929491500, −6.35588322515550822633654610946, −5.34224322482910341135394867103, −4.14601637374527638518628627132, −2.98423375672260046962489852045, −2.69599182736743100635042367762, 1.15455063562523356887372729626, 1.93823337165315958136070612136, 3.22724495349691775309205764549, 4.07111953783643336702309360647, 4.96030865523029737573896222912, 6.65274049394925123662916100065, 7.49922610721714755645109313918, 8.170062467679109937365387515830, 8.968839260361737352136230860030, 9.549284604149497947217750573232

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