Properties

Label 2-5070-13.12-c1-0-99
Degree $2$
Conductor $5070$
Sign $-0.246 - 0.969i$
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 − 3.44i·7-s + i·8-s + 9-s − 10-s − 4.24i·11-s − 12-s − 3.44·14-s i·15-s + 16-s − 6.78·17-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 − 1.30i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.28i·11-s − 0.288·12-s − 0.920·14-s − 0.258i·15-s + 0.250·16-s − 1.64·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.246 - 0.969i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5624765878\)
\(L(\frac12)\) \(\approx\) \(0.5624765878\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.44iT - 7T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
17 \( 1 + 6.78T + 17T^{2} \)
19 \( 1 - 6.26iT - 19T^{2} \)
23 \( 1 - 1.30T + 23T^{2} \)
29 \( 1 + 9.14T + 29T^{2} \)
31 \( 1 - 3.75iT - 31T^{2} \)
37 \( 1 + 6.82iT - 37T^{2} \)
41 \( 1 - 4.26iT - 41T^{2} \)
43 \( 1 + 3.07T + 43T^{2} \)
47 \( 1 + 7.76iT - 47T^{2} \)
53 \( 1 + 8.93T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 2.53T + 61T^{2} \)
67 \( 1 + 0.0760iT - 67T^{2} \)
71 \( 1 - 0.374iT - 71T^{2} \)
73 \( 1 - 16.7iT - 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 0.740iT - 83T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−7.81998404904056744857088585831, −7.21263601307506748894325141957, −6.25584553425688102067882840385, −5.41606495348098834504643395700, −4.41651198201354329779915296581, −3.83363572945920987749622210381, −3.29381609588877298272604138672, −2.09394379201956295066445672341, −1.24307363973396128094039139024, −0.13368404916328274429599631875, 1.96173845377314467885795007339, 2.44882212093264362702222610065, 3.46659737160920582675852300934, 4.59701464751868452104550860869, 4.94752695081701915989454958706, 6.04243152371806452487416290634, 6.67816207451123191627810901604, 7.25266339075493303420993551798, 7.971802823074793637043990248409, 8.777039633545669151737727377366

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