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Properties

Label 2-1050-5.2-c2-0-12
Degree $2$
Conductor $1050$
Sign $0.945 + 0.326i$
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 − 19.6·11-s + (−2.44 − 2.44i)12-s + (−0.747 + 0.747i)13-s + 3.74i·14-s − 4·16-s + (20.6 + 20.6i)17-s + (−2.99 + 2.99i)18-s − 13.7i·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 − 1.78·11-s + (−0.204 − 0.204i)12-s + (−0.0574 + 0.0574i)13-s + 0.267i·14-s − 0.250·16-s + (1.21 + 1.21i)17-s + (−0.166 + 0.166i)18-s − 0.721i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.945 + 0.326i$
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: \((2,\ 1050,\ (\ :1),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8347537758\)
\(L(\frac12)\) \(\approx\) \(0.8347537758\)
\(L(2)\) 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 \( 1 + 19.6T + 121T^{2} \)
13 \( 1 + (0.747 - 0.747i)T - 169iT^{2} \)
17 \( 1 + (-20.6 - 20.6i)T + 289iT^{2} \)
19 \( 1 + 13.7iT - 361T^{2} \)
23 \( 1 + (15.2 - 15.2i)T - 529iT^{2} \)
29 \( 1 + 52.0iT - 841T^{2} \)
31 \( 1 - 6.68T + 961T^{2} \)
37 \( 1 + (34.0 + 34.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 2.98T + 1.68e3T^{2} \)
43 \( 1 + (16.2 - 16.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-62.2 - 62.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-21.3 + 21.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 95.6iT - 3.48e3T^{2} \)
61 \( 1 + 31.9T + 3.72e3T^{2} \)
67 \( 1 + (-41.4 - 41.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 108.T + 5.04e3T^{2} \)
73 \( 1 + (-79.9 + 79.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 87.2iT - 6.24e3T^{2} \)
83 \( 1 + (17.2 - 17.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 60.6iT - 7.92e3T^{2} \)
97 \( 1 + (-49.0 - 49.0i)T + 9.40e3iT^{2} \)
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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

−10.03826607563062120529471232105, −9.011495837781891975369560532606, −7.936882474551648575634900001580, −7.53072343486918848006928124754, −6.14144419882289943154566577352, −5.39229227783854526276414583039, −4.27217671037214156789467702251, −3.29660428050365296534939068369, −2.19100286421403794814699162810, −0.57516907752235252399628029504, 0.62123331105063167275318555110, 2.16242485265301082084770871683, 3.30045904930697423820925178136, 5.18928713587367379104531861574, 5.30603783408655017864444867474, 6.50797974638386518499978263801, 7.33935668596094521624932510392, 7.998856453944803489674115798741, 8.736559352042101676419860903339, 9.992207174624559510709151378885

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