Properties

Label 2-1050-5.2-c2-0-35
Degree $2$
Conductor $1050$
Sign $-0.130 + 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.130 + 0.991i$
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: \((2,\ 1050,\ (\ :1),\ -0.130 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.367561164\)
\(L(\frac12)\) \(\approx\) \(1.367561164\)
\(L(2)\) 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

−9.271564372766634788717401562129, −8.448238809715563897827579946256, −7.77548004033462246939785448956, −7.03190826759645610786152959625, −6.13376851875567619883621229017, −5.07009555569761315622030198493, −4.47787662166048760851628961090, −2.85687468381133582332288005807, −2.40278516330522553583778945059, −0.30868803865896551725286423018, 1.63525431542195105391237405021, 2.72023888126736304881570115913, 3.65936777790417519352265573488, 4.67352930646998996895937991071, 5.35072970640850658451544918384, 6.42161983468541618609962454010, 7.62486758050364549426646844166, 8.288439441292010753426299148487, 9.243711788793009931850979682226, 10.12617547029012749150293792723

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