Properties

Label 2-1050-3.2-c2-0-70
Degree $2$
Conductor $1050$
Sign $-0.903 - 0.429i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (1.28 − 2.70i)3-s − 2.00·4-s + (−3.83 − 1.82i)6-s + 2.64·7-s + 2.82i·8-s + (−5.68 − 6.97i)9-s + 10.4i·11-s + (−2.57 + 5.41i)12-s + 11.9·13-s − 3.74i·14-s + 4.00·16-s − 29.1i·17-s + (−9.86 + 8.03i)18-s − 20.3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.429 − 0.903i)3-s − 0.500·4-s + (−0.638 − 0.303i)6-s + 0.377·7-s + 0.353i·8-s + (−0.631 − 0.775i)9-s + 0.951i·11-s + (−0.214 + 0.451i)12-s + 0.922·13-s − 0.267i·14-s + 0.250·16-s − 1.71i·17-s + (−0.548 + 0.446i)18-s − 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.903 - 0.429i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.903 - 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.343071467\)
\(L(\frac12)\) \(\approx\) \(1.343071467\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-1.28 + 2.70i)T \)
5 \( 1 \)
7 \( 1 - 2.64T \)
good11 \( 1 - 10.4iT - 121T^{2} \)
13 \( 1 - 11.9T + 169T^{2} \)
17 \( 1 + 29.1iT - 289T^{2} \)
19 \( 1 + 20.3T + 361T^{2} \)
23 \( 1 + 7.71iT - 529T^{2} \)
29 \( 1 + 47.4iT - 841T^{2} \)
31 \( 1 + 35.5T + 961T^{2} \)
37 \( 1 + 58.0T + 1.36e3T^{2} \)
41 \( 1 + 52.5iT - 1.68e3T^{2} \)
43 \( 1 + 15.7T + 1.84e3T^{2} \)
47 \( 1 - 85.2iT - 2.20e3T^{2} \)
53 \( 1 + 42.7iT - 2.80e3T^{2} \)
59 \( 1 - 37.9iT - 3.48e3T^{2} \)
61 \( 1 - 53.5T + 3.72e3T^{2} \)
67 \( 1 - 27.6T + 4.48e3T^{2} \)
71 \( 1 + 58.0iT - 5.04e3T^{2} \)
73 \( 1 + 67.8T + 5.32e3T^{2} \)
79 \( 1 + 19.2T + 6.24e3T^{2} \)
83 \( 1 + 33.6iT - 6.88e3T^{2} \)
89 \( 1 - 46.8iT - 7.92e3T^{2} \)
97 \( 1 - 65.1T + 9.40e3T^{2} \)
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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

−9.151895102817724020235674030540, −8.559296541871918249474801756155, −7.60726744320415167061304486689, −6.90310748743483039227493581671, −5.83527396428802806235223477327, −4.70426723415010664190521215833, −3.68410188427666135561454672453, −2.49312263036311919078511246101, −1.73184826248557980466993087196, −0.37968216723773434644849002511, 1.69446898098129363629481603343, 3.43195436803877725956964418739, 3.92969686212735197262146315208, 5.12838031411256752807986925168, 5.83164829179393374622249342640, 6.74835365336813136105699152674, 8.022813727097642661922419548187, 8.632600202702073569573638718008, 8.925907597850302583476220041523, 10.31349785792336450332121808976

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