Properties

Label 2-5070-13.12-c1-0-8
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 + 3.35i·7-s + i·8-s + 9-s + 10-s + 1.69i·11-s − 12-s + 3.35·14-s + i·15-s + 16-s − 0.939·17-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 + 1.26i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.510i·11-s − 0.288·12-s + 0.897·14-s + 0.258i·15-s + 0.250·16-s − 0.227·17-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

\(L(1)\) \(\approx\) \(0.9714908595\)
\(L(\frac12)\) \(\approx\) \(0.9714908595\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.35iT - 7T^{2} \)
11 \( 1 - 1.69iT - 11T^{2} \)
17 \( 1 + 0.939T + 17T^{2} \)
19 \( 1 - 4.85iT - 19T^{2} \)
23 \( 1 + 4.04T + 23T^{2} \)
29 \( 1 + 8.12T + 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 3.86iT - 41T^{2} \)
43 \( 1 + 4.02T + 43T^{2} \)
47 \( 1 + 1.27iT - 47T^{2} \)
53 \( 1 - 5.74T + 53T^{2} \)
59 \( 1 - 0.417iT - 59T^{2} \)
61 \( 1 - 0.198T + 61T^{2} \)
67 \( 1 + 8.93iT - 67T^{2} \)
71 \( 1 - 5.15iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 + 9.47iT - 89T^{2} \)
97 \( 1 + 15.4iT - 97T^{2} \)
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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

−8.549286225269346218772847717390, −7.984287324749540747528853188776, −7.22071829219101832923521291505, −6.18095820100433010674253816849, −5.62268400155058354299323338192, −4.66733921838803410133258803581, −3.79328883743510575398539333545, −3.09790316789841025519980116170, −2.18007226897392145921108419333, −1.73259479816555412221990929464, 0.23083537903928018218423522610, 1.31430819190766964790392523621, 2.57164567358161371527941690275, 3.82261613198440520883221864063, 4.05786913036622617794477921419, 5.07496434762529082265030403727, 5.77100497645270385689518570355, 6.82594083645211394892921101520, 7.19881232738701476589647172289, 7.957589744969236483224204726165

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