Properties

Label 2-5070-13.12-c1-0-56
Degree $2$
Conductor $5070$
Sign $0.999 + 0.0304i$
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 + 1.69i·7-s i·8-s + 9-s − 10-s + 4.55i·11-s + 12-s − 1.69·14-s i·15-s + 16-s − 2.35·17-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 + 0.639i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.37i·11-s + 0.288·12-s − 0.452·14-s − 0.258i·15-s + 0.250·16-s − 0.571·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.999 + 0.0304i$
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: \((2,\ 5070,\ (\ :1/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8160912384\)
\(L(\frac12)\) \(\approx\) \(0.8160912384\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69iT - 7T^{2} \)
11 \( 1 - 4.55iT - 11T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 + 6.51iT - 19T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 - 9.07T + 29T^{2} \)
31 \( 1 + 10.6iT - 31T^{2} \)
37 \( 1 + 6.18iT - 37T^{2} \)
41 \( 1 - 3.00iT - 41T^{2} \)
43 \( 1 - 4.93T + 43T^{2} \)
47 \( 1 - 4.28iT - 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + 4.32iT - 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 3.24iT - 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 - 6.72iT - 73T^{2} \)
79 \( 1 - 5.67T + 79T^{2} \)
83 \( 1 + 7.71iT - 83T^{2} \)
89 \( 1 - 9.12iT - 89T^{2} \)
97 \( 1 + 4.40iT - 97T^{2} \)
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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

−7.87422096569913188648727440239, −7.57282565500440446796273367690, −6.52774595925722188672868640611, −6.32016742724110061705281405754, −5.41753645083922910864165038063, −4.53837500518646834613308368130, −4.18217783411873223872517751340, −2.70181368618721728365467700752, −1.98485520790170391133114084274, −0.30340717774835447534491289212, 0.865356396450850205167519087393, 1.64974660127367491576258836081, 2.93487918029643805934258069774, 3.81180721996032855076429499071, 4.37681843133250048585484199756, 5.28252712175064401627976964208, 6.00581419016006499277032572703, 6.59321111774236030356562224266, 7.71067149517681271293787408442, 8.373155921936392215865095126674

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