Properties

Label 2-5070-1.1-c1-0-53
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 + 3.35·7-s + 8-s + 9-s − 10-s + 1.69·11-s + 12-s + 3.35·14-s − 15-s + 16-s + 0.939·17-s + 18-s − 4.85·19-s − 20-s + 3.35·21-s + 1.69·22-s + 4.04·23-s + 24-s + 25-s + 27-s + 3.35·28-s − 8.12·29-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 + 1.26·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.510·11-s + 0.288·12-s + 0.897·14-s − 0.258·15-s + 0.250·16-s + 0.227·17-s + 0.235·18-s − 1.11·19-s − 0.223·20-s + 0.732·21-s + 0.360·22-s + 0.844·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.634·28-s − 1.50·29-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

\(L(1)\) \(\approx\) \(4.547412178\)
\(L(\frac12)\) \(\approx\) \(4.547412178\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.35T + 7T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
17 \( 1 - 0.939T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 - 4.04T + 23T^{2} \)
29 \( 1 + 8.12T + 29T^{2} \)
31 \( 1 - 4.08T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 3.86T + 41T^{2} \)
43 \( 1 - 4.02T + 43T^{2} \)
47 \( 1 + 1.27T + 47T^{2} \)
53 \( 1 - 5.74T + 53T^{2} \)
59 \( 1 - 0.417T + 59T^{2} \)
61 \( 1 - 0.198T + 61T^{2} \)
67 \( 1 - 8.93T + 67T^{2} \)
71 \( 1 + 5.15T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.042090847558868501753061953150, −7.62903503615413829554150673602, −6.84439824012692477389042504955, −5.99916422967516959369248192442, −5.15176396055753321546959436186, −4.32024046025299993374760019575, −3.99766922502014349560981145629, −2.88220139527315090851176651076, −2.07214080818837533350997329564, −1.09726644194426819937869221615, 1.09726644194426819937869221615, 2.07214080818837533350997329564, 2.88220139527315090851176651076, 3.99766922502014349560981145629, 4.32024046025299993374760019575, 5.15176396055753321546959436186, 5.99916422967516959369248192442, 6.84439824012692477389042504955, 7.62903503615413829554150673602, 8.042090847558868501753061953150

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