Properties

Label 2-5070-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 4.60·7-s − 8-s + 9-s + 10-s − 12-s + 4.60·14-s + 15-s + 16-s − 4.60·17-s − 18-s + 4.60·19-s − 20-s + 4.60·21-s + 1.39·23-s + 24-s + 25-s − 27-s − 4.60·28-s + 4.60·29-s − 30-s − 6·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.74·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s + 1.23·14-s + 0.258·15-s + 0.250·16-s − 1.11·17-s − 0.235·18-s + 1.05·19-s − 0.223·20-s + 1.00·21-s + 0.290·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.870·28-s + 0.855·29-s − 0.182·30-s − 1.07·31-s − 0.176·32-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: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2936254812\)
\(L(\frac12)\) \(\approx\) \(0.2936254812\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.60T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 - 4.60T + 19T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 - 4.60T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 9.21T + 37T^{2} \)
41 \( 1 + 3.21T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 3.21T + 67T^{2} \)
71 \( 1 + 9.21T + 71T^{2} \)
73 \( 1 + 1.39T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + 1.39T + 97T^{2} \)
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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

−8.323620248736351660196934698873, −7.28151780679887243036468312402, −6.87802941084421675318324625514, −6.31632431519125725847879047382, −5.49097569526049405539048209607, −4.57087270893804603425847329871, −3.45626900019467871050443685003, −3.00659524833967304745832863791, −1.66874533278296205482601827815, −0.33094778280911462903471142518, 0.33094778280911462903471142518, 1.66874533278296205482601827815, 3.00659524833967304745832863791, 3.45626900019467871050443685003, 4.57087270893804603425847329871, 5.49097569526049405539048209607, 6.31632431519125725847879047382, 6.87802941084421675318324625514, 7.28151780679887243036468312402, 8.323620248736351660196934698873

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