Properties

Label 2-5070-13.12-c1-0-48
Degree $2$
Conductor $5070$
Sign $0.691 + 0.722i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + 3-s − 4-s i·5-s i·6-s − 1.75i·7-s + i·8-s + 9-s − 10-s − 1.19i·11-s − 12-s − 1.75·14-s i·15-s + 16-s + 6.76·17-s i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.662i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.361i·11-s − 0.288·12-s − 0.468·14-s − 0.258i·15-s + 0.250·16-s + 1.64·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.531758840\)
\(L(\frac12)\) \(\approx\) \(2.531758840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 + 1.19iT - 11T^{2} \)
17 \( 1 - 6.76T + 17T^{2} \)
19 \( 1 - 8.63iT - 19T^{2} \)
23 \( 1 - 6.04T + 23T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 - 6.80iT - 31T^{2} \)
37 \( 1 - 4.34iT - 37T^{2} \)
41 \( 1 - 6.63iT - 41T^{2} \)
43 \( 1 + 4.07T + 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + 13.7iT - 59T^{2} \)
61 \( 1 + 1.18T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 + 1.59iT - 73T^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
83 \( 1 - 8.40iT - 83T^{2} \)
89 \( 1 + 4.49iT - 89T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−8.162139766708328005131381227726, −7.75551774705680030929924503061, −6.78904837768730958776016128326, −5.82409169619316438657609604269, −5.04936534958229252930142728427, −4.26458949944005174961018015484, −3.37317424653556137921514007711, −3.01903863790114294665378567774, −1.51037591269071385119672052316, −1.07481664061439415025040204479, 0.77554403893384420728337870847, 2.25270437277617276257157482121, 2.96564075642220655788875884893, 3.79966808323726158849833781461, 4.84168437194574718962993812089, 5.39166804810466838771265038870, 6.25285386716315589201985654803, 7.12785222183625608529665987711, 7.40525833281566744486141942373, 8.308062445000546880053781041905

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