Properties

Label 2-5070-1.1-c1-0-42
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 − 3.04·7-s − 8-s + 9-s + 10-s − 1.13·11-s − 12-s + 3.04·14-s + 15-s + 16-s + 2.69·17-s − 18-s − 1.55·19-s − 20-s + 3.04·21-s + 1.13·22-s + 5.40·23-s + 24-s + 25-s − 27-s − 3.04·28-s − 6.15·29-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 − 1.15·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.342·11-s − 0.288·12-s + 0.814·14-s + 0.258·15-s + 0.250·16-s + 0.652·17-s − 0.235·18-s − 0.356·19-s − 0.223·20-s + 0.665·21-s + 0.242·22-s + 1.12·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.576·28-s − 1.14·29-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 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 + 1.13T + 11T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 - 5.40T + 23T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 0.246T + 31T^{2} \)
37 \( 1 - 0.554T + 37T^{2} \)
41 \( 1 + 8.26T + 41T^{2} \)
43 \( 1 - 2.03T + 43T^{2} \)
47 \( 1 - 6.70T + 47T^{2} \)
53 \( 1 - 5.77T + 53T^{2} \)
59 \( 1 + 1.36T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 - 8.57T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 - 5.33T + 79T^{2} \)
83 \( 1 - 5.45T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 1.93T + 97T^{2} \)
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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

−7.893531403007937220462835591154, −6.97832734829668655832231845459, −6.75446063912457778681237804636, −5.73143900944716364369844551479, −5.17818296966509723263788229667, −3.96452721617548886795754718796, −3.29024831008077842920425188978, −2.33159237780172804737933699493, −0.991237008844185505469845454745, 0, 0.991237008844185505469845454745, 2.33159237780172804737933699493, 3.29024831008077842920425188978, 3.96452721617548886795754718796, 5.17818296966509723263788229667, 5.73143900944716364369844551479, 6.75446063912457778681237804636, 6.97832734829668655832231845459, 7.893531403007937220462835591154

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