Properties

Label 2-847-11.4-c1-0-47
Degree $2$
Conductor $847$
Sign $0.530 + 0.847i$
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  + (1.60 + 1.16i)2-s + (0.861 − 2.65i)3-s + (0.598 + 1.84i)4-s + (0.0217 − 0.0158i)5-s + (4.47 − 3.25i)6-s + (−0.309 − 0.951i)7-s + (0.0378 − 0.116i)8-s + (−3.85 − 2.80i)9-s + 0.0534·10-s + 5.40·12-s + (−3.94 − 2.86i)13-s + (0.613 − 1.88i)14-s + (−0.0231 − 0.0713i)15-s + (3.33 − 2.42i)16-s + (1.35 − 0.986i)17-s + (−2.92 − 8.99i)18-s + ⋯
L(s)  = 1  + (1.13 + 0.824i)2-s + (0.497 − 1.52i)3-s + (0.299 + 0.921i)4-s + (0.00974 − 0.00707i)5-s + (1.82 − 1.32i)6-s + (−0.116 − 0.359i)7-s + (0.0133 − 0.0411i)8-s + (−1.28 − 0.933i)9-s + 0.0168·10-s + 1.55·12-s + (−1.09 − 0.795i)13-s + (0.163 − 0.504i)14-s + (−0.00598 − 0.0184i)15-s + (0.833 − 0.605i)16-s + (0.329 − 0.239i)17-s + (−0.688 − 2.11i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72796 - 1.51163i\)
\(L(\frac12)\) \(\approx\) \(2.72796 - 1.51163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-1.60 - 1.16i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.861 + 2.65i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.0217 + 0.0158i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.94 + 2.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.35 + 0.986i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.424 - 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.06T + 23T^{2} \)
29 \( 1 + (-1.97 - 6.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.24 + 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.161 - 0.495i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + (2.76 - 8.52i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.20 - 2.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.00 - 9.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.85 + 4.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + (1.65 - 1.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.23 - 9.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.73 - 3.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.10 + 1.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.21T + 89T^{2} \)
97 \( 1 + (2.74 + 1.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.979055543404276209816026598002, −8.904732135172185865089067472906, −7.78876559467907028845620042062, −7.25593105932844950607635647104, −6.82346300128460013127356830782, −5.69364573235757068404080049343, −5.00544687049541223310916033304, −3.56901800770701035762435204324, −2.66099980142861213889378784597, −1.06160291079755931323007400937, 2.26250271788726130250397360202, 3.06224868869202291979414767360, 3.97298114465773764747020350235, 4.76974721966623913320213004473, 5.26052521953940813523837071273, 6.58419295138271151521776799560, 8.039259308768816550067908676233, 8.948389403396463128014701267520, 9.745625185556281926411834571932, 10.33251627495638698662323093551

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