Properties

Label 2-8470-1.1-c1-0-114
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s − 6·13-s + 14-s + 16-s + 6·17-s + 3·18-s + 6·19-s − 20-s + 8·23-s + 25-s + 6·26-s − 28-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 35-s − 3·36-s − 4·37-s − 6·38-s + 40-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 1.37·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s − 1/2·36-s − 0.657·37-s − 0.973·38-s + 0.158·40-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: yes
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.47275411963440337794882943605, −7.16625091924132139551037508014, −6.06002593200817576417677308102, −5.36332490350365140438107205580, −4.89962138351509561722300789879, −3.42636580384166541105292756242, −3.17088884485128297302019586200, −2.23242126154818294190934771949, −0.992445695085263760308742122746, 0, 0.992445695085263760308742122746, 2.23242126154818294190934771949, 3.17088884485128297302019586200, 3.42636580384166541105292756242, 4.89962138351509561722300789879, 5.36332490350365140438107205580, 6.06002593200817576417677308102, 7.16625091924132139551037508014, 7.47275411963440337794882943605

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