Properties

Label 2-5070-13.12-c1-0-24
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 − 3.04i·7-s i·8-s + 9-s − 10-s − 1.13i·11-s + 12-s + 3.04·14-s i·15-s + 16-s − 2.69·17-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 − 1.15i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.342i·11-s + 0.288·12-s + 0.814·14-s − 0.258i·15-s + 0.250·16-s − 0.652·17-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

\(L(1)\) \(\approx\) \(1.013384847\)
\(L(\frac12)\) \(\approx\) \(1.013384847\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.04iT - 7T^{2} \)
11 \( 1 + 1.13iT - 11T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 1.55iT - 19T^{2} \)
23 \( 1 + 5.40T + 23T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 + 0.246iT - 31T^{2} \)
37 \( 1 - 0.554iT - 37T^{2} \)
41 \( 1 - 8.26iT - 41T^{2} \)
43 \( 1 + 2.03T + 43T^{2} \)
47 \( 1 - 6.70iT - 47T^{2} \)
53 \( 1 - 5.77T + 53T^{2} \)
59 \( 1 + 1.36iT - 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 2.47iT - 67T^{2} \)
71 \( 1 + 8.57iT - 71T^{2} \)
73 \( 1 - 14.8iT - 73T^{2} \)
79 \( 1 - 5.33T + 79T^{2} \)
83 \( 1 + 5.45iT - 83T^{2} \)
89 \( 1 + 16.8iT - 89T^{2} \)
97 \( 1 - 1.93iT - 97T^{2} \)
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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

−8.114263465453249525867603595683, −7.53752684091129744947990244832, −6.93298616429173893920606194878, −6.24998407789623265923805909975, −5.70951838807116753231821811839, −4.70267618663335665094006230453, −4.06911403856038903403681610501, −3.35504845592714652108366990047, −1.94618358620296586297386764952, −0.68389633364134249059480563018, 0.45154405976837365496967814657, 1.89392803709015107334140406139, 2.33707369050268072767400543576, 3.63252999630653618805458313009, 4.33257777362928464317868344958, 5.27366552222166158009247037677, 5.60784030012467952149453910987, 6.52297014861590520803285429172, 7.38411864789853440506407852471, 8.314162379047258734504016438795

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