Properties

Label 2-8470-1.1-c1-0-113
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.80·3-s + 4-s + 5-s + 2.80·6-s − 7-s − 8-s + 4.88·9-s − 10-s − 2.80·12-s + 3.05·13-s + 14-s − 2.80·15-s + 16-s − 6.94·17-s − 4.88·18-s + 4.69·19-s + 20-s + 2.80·21-s − 1.73·23-s + 2.80·24-s + 25-s − 3.05·26-s − 5.29·27-s − 28-s + 6.74·29-s + 2.80·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.62·3-s + 0.5·4-s + 0.447·5-s + 1.14·6-s − 0.377·7-s − 0.353·8-s + 1.62·9-s − 0.316·10-s − 0.810·12-s + 0.848·13-s + 0.267·14-s − 0.725·15-s + 0.250·16-s − 1.68·17-s − 1.15·18-s + 1.07·19-s + 0.223·20-s + 0.612·21-s − 0.360·23-s + 0.573·24-s + 0.200·25-s − 0.599·26-s − 1.01·27-s − 0.188·28-s + 1.25·29-s + 0.512·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(67.6332\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8470} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2.80T + 3T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 - 6.74T + 29T^{2} \)
31 \( 1 - 3.34T + 31T^{2} \)
37 \( 1 + 7.32T + 37T^{2} \)
41 \( 1 + 7.58T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 - 4.00T + 47T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 - 7.64T + 59T^{2} \)
61 \( 1 + 8.93T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 2.30T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + 4.25T + 79T^{2} \)
83 \( 1 + 3.86T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 6.94T + 97T^{2} \)
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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

−7.10597827977165386597989507373, −6.62501015972227914675189380272, −6.30718110768756257552554975522, −5.45907684171016062635985476484, −4.92578321849971740300159865175, −4.00284109873171763087394173816, −2.95984569850982958694848089400, −1.83893521082177798890848477504, −0.982979967444681421269924550978, 0, 0.982979967444681421269924550978, 1.83893521082177798890848477504, 2.95984569850982958694848089400, 4.00284109873171763087394173816, 4.92578321849971740300159865175, 5.45907684171016062635985476484, 6.30718110768756257552554975522, 6.62501015972227914675189380272, 7.10597827977165386597989507373

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