Properties

Label 2-5070-1.1-c1-0-100
Degree $2$
Conductor $5070$
Sign $-1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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 + 3-s + 4-s + 5-s + 6-s − 3.44·7-s + 8-s + 9-s + 10-s − 4.24·11-s + 12-s − 3.44·14-s + 15-s + 16-s + 6.78·17-s + 18-s − 6.26·19-s + 20-s − 3.44·21-s − 4.24·22-s − 1.30·23-s + 24-s + 25-s + 27-s − 3.44·28-s − 9.14·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.30·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.28·11-s + 0.288·12-s − 0.920·14-s + 0.258·15-s + 0.250·16-s + 1.64·17-s + 0.235·18-s − 1.43·19-s + 0.223·20-s − 0.751·21-s − 0.905·22-s − 0.272·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.651·28-s − 1.69·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
17 \( 1 - 6.78T + 17T^{2} \)
19 \( 1 + 6.26T + 19T^{2} \)
23 \( 1 + 1.30T + 23T^{2} \)
29 \( 1 + 9.14T + 29T^{2} \)
31 \( 1 + 3.75T + 31T^{2} \)
37 \( 1 + 6.82T + 37T^{2} \)
41 \( 1 + 4.26T + 41T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 + 7.76T + 47T^{2} \)
53 \( 1 + 8.93T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 2.53T + 61T^{2} \)
67 \( 1 - 0.0760T + 67T^{2} \)
71 \( 1 + 0.374T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 - 0.740T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.82015833392269467668766666903, −7.06411478265656074611572668885, −6.37568683958646331445120167253, −5.59946615019631795524472713991, −5.10402846570306680895657443090, −3.84137696285683360116665147451, −3.37766818113423552493457463966, −2.57317974669386789834925724601, −1.76248174778056944704193522496, 0, 1.76248174778056944704193522496, 2.57317974669386789834925724601, 3.37766818113423552493457463966, 3.84137696285683360116665147451, 5.10402846570306680895657443090, 5.59946615019631795524472713991, 6.37568683958646331445120167253, 7.06411478265656074611572668885, 7.82015833392269467668766666903

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