Properties

Label 2-8470-1.1-c1-0-198
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 + 0.414·3-s + 4-s + 5-s + 0.414·6-s − 7-s + 8-s − 2.82·9-s + 10-s + 0.414·12-s + 0.696·13-s − 14-s + 0.414·15-s + 16-s − 0.449·17-s − 2.82·18-s − 3.37·19-s + 20-s − 0.414·21-s − 9.00·23-s + 0.414·24-s + 25-s + 0.696·26-s − 2.41·27-s − 28-s − 1.60·29-s + 0.414·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s − 0.377·7-s + 0.353·8-s − 0.942·9-s + 0.316·10-s + 0.119·12-s + 0.193·13-s − 0.267·14-s + 0.106·15-s + 0.250·16-s − 0.109·17-s − 0.666·18-s − 0.775·19-s + 0.223·20-s − 0.0903·21-s − 1.87·23-s + 0.0845·24-s + 0.200·25-s + 0.136·26-s − 0.464·27-s − 0.188·28-s − 0.297·29-s + 0.0756·30-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 \( 1 - 0.414T + 3T^{2} \)
13 \( 1 - 0.696T + 13T^{2} \)
17 \( 1 + 0.449T + 17T^{2} \)
19 \( 1 + 3.37T + 19T^{2} \)
23 \( 1 + 9.00T + 23T^{2} \)
29 \( 1 + 1.60T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 - 6.12T + 41T^{2} \)
43 \( 1 - 7.14T + 43T^{2} \)
47 \( 1 + 9.51T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 + 8.29T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 0.585T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 8.56T + 97T^{2} \)
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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

−7.41957852896298728476344021586, −6.34966342145183934889681302206, −6.14820409278514930508190411025, −5.51661880797613135613941570897, −4.47719188375516881382919912542, −3.99895721646085089034625954409, −2.92564037308280975596712915890, −2.52495436365313852298061959061, −1.51433848791891405573202105470, 0, 1.51433848791891405573202105470, 2.52495436365313852298061959061, 2.92564037308280975596712915890, 3.99895721646085089034625954409, 4.47719188375516881382919912542, 5.51661880797613135613941570897, 6.14820409278514930508190411025, 6.34966342145183934889681302206, 7.41957852896298728476344021586

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