Properties

Label 2-8470-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.90·3-s + 4-s − 5-s − 2.90·6-s − 7-s + 8-s + 5.43·9-s − 10-s − 2.90·12-s − 6.51·13-s − 14-s + 2.90·15-s + 16-s + 7.07·17-s + 5.43·18-s + 3.97·19-s − 20-s + 2.90·21-s + 9.39·23-s − 2.90·24-s + 25-s − 6.51·26-s − 7.08·27-s − 28-s + 1.17·29-s + 2.90·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.447·5-s − 1.18·6-s − 0.377·7-s + 0.353·8-s + 1.81·9-s − 0.316·10-s − 0.838·12-s − 1.80·13-s − 0.267·14-s + 0.750·15-s + 0.250·16-s + 1.71·17-s + 1.28·18-s + 0.911·19-s − 0.223·20-s + 0.633·21-s + 1.96·23-s − 0.592·24-s + 0.200·25-s − 1.27·26-s − 1.36·27-s − 0.188·28-s + 0.218·29-s + 0.530·30-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: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.264682464\)
\(L(\frac12)\) \(\approx\) \(1.264682464\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.90T + 3T^{2} \)
13 \( 1 + 6.51T + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 - 3.97T + 19T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 1.28T + 31T^{2} \)
37 \( 1 + 1.00T + 37T^{2} \)
41 \( 1 + 5.43T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 7.75T + 53T^{2} \)
59 \( 1 + 5.64T + 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 + 0.259T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 + 1.43T + 89T^{2} \)
97 \( 1 + 8.60T + 97T^{2} \)
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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

−7.43416444364358356332924726493, −6.90068880327784776471880477278, −6.44223286243658457518604590662, −5.33003226411573674545156208102, −5.16495966679951775556950381571, −4.70230323888170165572458639876, −3.46892130123215579770648941987, −2.99062239401367938632965236246, −1.54301787602735412524392174222, −0.55923809348416851371711986409, 0.55923809348416851371711986409, 1.54301787602735412524392174222, 2.99062239401367938632965236246, 3.46892130123215579770648941987, 4.70230323888170165572458639876, 5.16495966679951775556950381571, 5.33003226411573674545156208102, 6.44223286243658457518604590662, 6.90068880327784776471880477278, 7.43416444364358356332924726493

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