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Properties

Label 2-1050-5.3-c2-0-18
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 − 3.32·11-s + (2.44 − 2.44i)12-s + (7.80 + 7.80i)13-s − 3.74i·14-s − 4·16-s + (13.6 − 13.6i)17-s + (2.99 + 2.99i)18-s + 37.1i·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 − 0.302·11-s + (0.204 − 0.204i)12-s + (0.600 + 0.600i)13-s − 0.267i·14-s − 0.250·16-s + (0.803 − 0.803i)17-s + (0.166 + 0.166i)18-s + 1.95i·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

\(L(\frac{3}{2})\) \(\approx\) \(3.177340026\)
\(L(\frac12)\) \(\approx\) \(3.177340026\)
\(L(2)\) 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 \( 1 + 3.32T + 121T^{2} \)
13 \( 1 + (-7.80 - 7.80i)T + 169iT^{2} \)
17 \( 1 + (-13.6 + 13.6i)T - 289iT^{2} \)
19 \( 1 - 37.1iT - 361T^{2} \)
23 \( 1 + (-31.5 - 31.5i)T + 529iT^{2} \)
29 \( 1 + 55.3iT - 841T^{2} \)
31 \( 1 + 4.54T + 961T^{2} \)
37 \( 1 + (-46.8 + 46.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 9.94T + 1.68e3T^{2} \)
43 \( 1 + (23.0 + 23.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-6.38 + 6.38i)T - 2.20e3iT^{2} \)
53 \( 1 + (-42.9 - 42.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 59.2iT - 3.48e3T^{2} \)
61 \( 1 - 47.1T + 3.72e3T^{2} \)
67 \( 1 + (8.48 - 8.48i)T - 4.48e3iT^{2} \)
71 \( 1 - 85.6T + 5.04e3T^{2} \)
73 \( 1 + (-34.7 - 34.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 96.4iT - 6.24e3T^{2} \)
83 \( 1 + (19.6 + 19.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + (88.5 - 88.5i)T - 9.40e3iT^{2} \)
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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

−9.772152899872948472562481299550, −9.110951831727962410869562337355, −7.984158625891118257799452556188, −7.34131508644018931866073640319, −5.97268131627348975008723936536, −5.30864442083401273816134787019, −4.14085605994639111646867448668, −3.55609826610672485170653324538, −2.36565414172504673291639838042, −1.13695215194213739784287382944, 0.997581019110429314560062639061, 2.60815872989478718707888887263, 3.34497605679330052864587350930, 4.69504207523138517178353805481, 5.38575926857879662394666409397, 6.53046907554850923128287565295, 7.06944163220245616316421499799, 8.226788134096685170406916701269, 8.546304933102567153712791530828, 9.538345880534704927708410274465

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