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Properties

Label 2-1050-15.14-c2-0-17
Degree $2$
Conductor $1050$
Sign $-0.700 - 0.713i$
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  + 1.41·2-s + (0.922 + 2.85i)3-s + 2.00·4-s + (1.30 + 4.03i)6-s − 2.64i·7-s + 2.82·8-s + (−7.29 + 5.26i)9-s + 4.58i·11-s + (1.84 + 5.70i)12-s + 20.4i·13-s − 3.74i·14-s + 4.00·16-s − 4.97·17-s + (−10.3 + 7.45i)18-s − 6.22·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.307 + 0.951i)3-s + 0.500·4-s + (0.217 + 0.672i)6-s − 0.377i·7-s + 0.353·8-s + (−0.810 + 0.585i)9-s + 0.416i·11-s + (0.153 + 0.475i)12-s + 1.57i·13-s − 0.267i·14-s + 0.250·16-s − 0.292·17-s + (−0.573 + 0.413i)18-s − 0.327·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.569201253\)
\(L(\frac12)\) \(\approx\) \(2.569201253\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-0.922 - 2.85i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 4.58iT - 121T^{2} \)
13 \( 1 - 20.4iT - 169T^{2} \)
17 \( 1 + 4.97T + 289T^{2} \)
19 \( 1 + 6.22T + 361T^{2} \)
23 \( 1 - 0.140T + 529T^{2} \)
29 \( 1 - 1.76iT - 841T^{2} \)
31 \( 1 + 29.7T + 961T^{2} \)
37 \( 1 - 38.8iT - 1.36e3T^{2} \)
41 \( 1 - 68.8iT - 1.68e3T^{2} \)
43 \( 1 + 20.3iT - 1.84e3T^{2} \)
47 \( 1 - 57.3T + 2.20e3T^{2} \)
53 \( 1 - 41.0T + 2.80e3T^{2} \)
59 \( 1 - 79.4iT - 3.48e3T^{2} \)
61 \( 1 + 19.1T + 3.72e3T^{2} \)
67 \( 1 - 67.5iT - 4.48e3T^{2} \)
71 \( 1 + 30.8iT - 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 - 20.9T + 6.24e3T^{2} \)
83 \( 1 + 156.T + 6.88e3T^{2} \)
89 \( 1 - 133. iT - 7.92e3T^{2} \)
97 \( 1 + 20.1iT - 9.40e3T^{2} \)
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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

−10.05651635995802659553698295695, −9.287645211206465571308527732411, −8.517694146707297394796814559712, −7.39350787130861657266592630341, −6.59086189843584182135261112365, −5.55462750711613814742496951381, −4.45809729983318896308603285060, −4.15241839921622430558454884912, −2.95383984585513286792124619986, −1.81768834543972404741620268746, 0.55684801066376145096960891635, 2.05109794315442654091597558054, 2.95164566447675346755551989463, 3.87801417043942794475394166610, 5.42448576122329699929774609204, 5.81156952510281917098945351711, 6.87716716316964209369236219369, 7.64714992994338192342878034560, 8.417043989155300502308459661194, 9.207334906691698287667953441702

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