Properties

Label 2-8470-1.1-c1-0-0
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 − 0.732·3-s + 4-s − 5-s + 0.732·6-s − 7-s − 8-s − 2.46·9-s + 10-s − 0.732·12-s − 5.46·13-s + 14-s + 0.732·15-s + 16-s − 3.46·17-s + 2.46·18-s − 3.26·19-s − 20-s + 0.732·21-s + 2.19·23-s + 0.732·24-s + 25-s + 5.46·26-s + 4·27-s − 28-s + 1.26·29-s − 0.732·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s − 0.377·7-s − 0.353·8-s − 0.821·9-s + 0.316·10-s − 0.211·12-s − 1.51·13-s + 0.267·14-s + 0.189·15-s + 0.250·16-s − 0.840·17-s + 0.580·18-s − 0.749·19-s − 0.223·20-s + 0.159·21-s + 0.457·23-s + 0.149·24-s + 0.200·25-s + 1.07·26-s + 0.769·27-s − 0.188·28-s + 0.235·29-s − 0.133·30-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

\(L(1)\) \(\approx\) \(0.07326858123\)
\(L(\frac12)\) \(\approx\) \(0.07326858123\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 8.19T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + 6.39T + 73T^{2} \)
79 \( 1 - 1.80T + 79T^{2} \)
83 \( 1 + 4.39T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−7.962022869841743540773668008923, −6.91293904617056177845845610519, −6.71941525247226196483818943579, −5.84157406733271623585095954346, −4.97904534958620749140701463199, −4.43723244118109288391690677305, −3.19437633866738182789122358667, −2.67997533242755250837786509196, −1.65609134455063738807448029125, −0.14404834769734862854217692458, 0.14404834769734862854217692458, 1.65609134455063738807448029125, 2.67997533242755250837786509196, 3.19437633866738182789122358667, 4.43723244118109288391690677305, 4.97904534958620749140701463199, 5.84157406733271623585095954346, 6.71941525247226196483818943579, 6.91293904617056177845845610519, 7.962022869841743540773668008923

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