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Properties

Label 2-1050-5.3-c2-0-13
Degree $2$
Conductor $1050$
Sign $0.973 + 0.229i$
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 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + 11.5·11-s + (−2.44 + 2.44i)12-s + (2.01 + 2.01i)13-s + 3.74i·14-s − 4·16-s + (−7.75 + 7.75i)17-s + (2.99 + 2.99i)18-s + 21.8i·19-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + 1.05·11-s + (−0.204 + 0.204i)12-s + (0.155 + 0.155i)13-s + 0.267i·14-s − 0.250·16-s + (−0.456 + 0.456i)17-s + (0.166 + 0.166i)18-s + 1.14i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.052267094\)
\(L(\frac12)\) \(\approx\) \(2.052267094\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 - 11.5T + 121T^{2} \)
13 \( 1 + (-2.01 - 2.01i)T + 169iT^{2} \)
17 \( 1 + (7.75 - 7.75i)T - 289iT^{2} \)
19 \( 1 - 21.8iT - 361T^{2} \)
23 \( 1 + (-7.41 - 7.41i)T + 529iT^{2} \)
29 \( 1 - 31.0iT - 841T^{2} \)
31 \( 1 + 11.5T + 961T^{2} \)
37 \( 1 + (-32.0 + 32.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 39.6T + 1.68e3T^{2} \)
43 \( 1 + (-19.3 - 19.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (21.8 - 21.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-42.4 - 42.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 89.0iT - 3.48e3T^{2} \)
61 \( 1 - 7.07T + 3.72e3T^{2} \)
67 \( 1 + (15.6 - 15.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 133.T + 5.04e3T^{2} \)
73 \( 1 + (-92.3 - 92.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-30.0 - 30.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 4.93iT - 7.92e3T^{2} \)
97 \( 1 + (-30.2 + 30.2i)T - 9.40e3iT^{2} \)
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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

−9.687366841262405085190843539744, −9.074924959713222239712280522982, −8.006769416649127616220495758335, −6.91245225118412500254063664361, −6.19235109308437354575982989516, −5.47431554922501982546556741635, −4.29074247978752908043045331724, −3.47509302115895941325003263796, −2.12704974801305659253434037333, −1.09486753129545131651301411338, 0.67866025824819209210389933579, 2.57925267446985995629792274292, 3.80204927834686459886381050596, 4.49369577068336199038961484599, 5.42798965158547884505246331550, 6.45738446528299634651572003307, 6.89259181152290231290256596240, 7.993514932244128608871923819017, 9.059989403944969849906697218232, 9.559318848333616734988100207555

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