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Properties

Label 2-1050-5.2-c2-0-23
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: $\chi_{1050} (757, \cdot )$
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

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

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