Properties

Label 2-5070-1.1-c1-0-15
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 − 2.32·7-s − 8-s + 9-s − 10-s − 5.34·11-s + 12-s + 2.32·14-s + 15-s + 16-s + 4·17-s − 18-s + 4.02·19-s + 20-s − 2.32·21-s + 5.34·22-s − 4.93·23-s − 24-s + 25-s + 27-s − 2.32·28-s + 4.29·29-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 − 0.877·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.61·11-s + 0.288·12-s + 0.620·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 0.922·19-s + 0.223·20-s − 0.506·21-s + 1.13·22-s − 1.02·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.438·28-s + 0.796·29-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: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435515391\)
\(L(\frac12)\) \(\approx\) \(1.435515391\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 4.02T + 19T^{2} \)
23 \( 1 + 4.93T + 23T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 - 3.14T + 37T^{2} \)
41 \( 1 - 2.64T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 1.81T + 47T^{2} \)
53 \( 1 + 5.48T + 53T^{2} \)
59 \( 1 - 6.78T + 59T^{2} \)
61 \( 1 - 0.535T + 61T^{2} \)
67 \( 1 - 4.10T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−8.108173813203218789141526477278, −7.68652740971404147889208487382, −7.07017095533010779359126805737, −5.99201914243998547241336205304, −5.61858980325906754675658191968, −4.52380280267014941555016734959, −3.29872521028188318056054497034, −2.83985346275550562577635992449, −1.96546230718536684133851008700, −0.68609541638473039354441200815, 0.68609541638473039354441200815, 1.96546230718536684133851008700, 2.83985346275550562577635992449, 3.29872521028188318056054497034, 4.52380280267014941555016734959, 5.61858980325906754675658191968, 5.99201914243998547241336205304, 7.07017095533010779359126805737, 7.68652740971404147889208487382, 8.108173813203218789141526477278

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