Properties

Label 2-2070-1.1-c3-0-68
Degree $2$
Conductor $2070$
Sign $-1$
Analytic cond. $122.133$
Root an. cond. $11.0514$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 5·5-s − 18·7-s − 8·8-s − 10·10-s + 32·11-s − 47·13-s + 36·14-s + 16·16-s − 20·17-s + 36·19-s + 20·20-s − 64·22-s + 23·23-s + 25·25-s + 94·26-s − 72·28-s + 27·29-s − 33·31-s − 32·32-s + 40·34-s − 90·35-s + 56·37-s − 72·38-s − 40·40-s + 157·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.971·7-s − 0.353·8-s − 0.316·10-s + 0.877·11-s − 1.00·13-s + 0.687·14-s + 1/4·16-s − 0.285·17-s + 0.434·19-s + 0.223·20-s − 0.620·22-s + 0.208·23-s + 1/5·25-s + 0.709·26-s − 0.485·28-s + 0.172·29-s − 0.191·31-s − 0.176·32-s + 0.201·34-s − 0.434·35-s + 0.248·37-s − 0.307·38-s − 0.158·40-s + 0.598·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(122.133\)
Root analytic conductor: \(11.0514\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2070,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
5 \( 1 - p T \)
23 \( 1 - p T \)
good7 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 + 47 T + p^{3} T^{2} \)
17 \( 1 + 20 T + p^{3} T^{2} \)
19 \( 1 - 36 T + p^{3} T^{2} \)
29 \( 1 - 27 T + p^{3} T^{2} \)
31 \( 1 + 33 T + p^{3} T^{2} \)
37 \( 1 - 56 T + p^{3} T^{2} \)
41 \( 1 - 157 T + p^{3} T^{2} \)
43 \( 1 - 18 T + p^{3} T^{2} \)
47 \( 1 + 65 T + p^{3} T^{2} \)
53 \( 1 - 14 T + p^{3} T^{2} \)
59 \( 1 - 744 T + p^{3} T^{2} \)
61 \( 1 - 552 T + p^{3} T^{2} \)
67 \( 1 + 156 T + p^{3} T^{2} \)
71 \( 1 + 699 T + p^{3} T^{2} \)
73 \( 1 + 609 T + p^{3} T^{2} \)
79 \( 1 + 644 T + p^{3} T^{2} \)
83 \( 1 + 512 T + p^{3} T^{2} \)
89 \( 1 - 102 T + p^{3} T^{2} \)
97 \( 1 - 578 T + p^{3} T^{2} \)
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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

−8.608246176084150298496602396082, −7.49256360394649991840305465151, −6.88452048452857143759825598061, −6.21830338637405348188622606831, −5.35142508102068939697805029405, −4.20349226170209957092510642580, −3.13439089135123642125059341553, −2.29798757226791552332434209526, −1.13345102020249856276648043416, 0, 1.13345102020249856276648043416, 2.29798757226791552332434209526, 3.13439089135123642125059341553, 4.20349226170209957092510642580, 5.35142508102068939697805029405, 6.21830338637405348188622606831, 6.88452048452857143759825598061, 7.49256360394649991840305465151, 8.608246176084150298496602396082

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