Properties

Label 2-1050-3.2-c2-0-60
Degree $2$
Conductor $1050$
Sign $-0.961 + 0.273i$
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  − 1.41i·2-s + (−0.821 − 2.88i)3-s − 2.00·4-s + (−4.08 + 1.16i)6-s + 2.64·7-s + 2.82i·8-s + (−7.64 + 4.74i)9-s − 18.1i·11-s + (1.64 + 5.77i)12-s + 22.5·13-s − 3.74i·14-s + 4.00·16-s + 0.457i·17-s + (6.70 + 10.8i)18-s + 30.1·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.273 − 0.961i)3-s − 0.500·4-s + (−0.680 + 0.193i)6-s + 0.377·7-s + 0.353i·8-s + (−0.849 + 0.526i)9-s − 1.65i·11-s + (0.136 + 0.480i)12-s + 1.73·13-s − 0.267i·14-s + 0.250·16-s + 0.0268i·17-s + (0.372 + 0.600i)18-s + 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.961 + 0.273i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.763209639\)
\(L(\frac12)\) \(\approx\) \(1.763209639\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (0.821 + 2.88i)T \)
5 \( 1 \)
7 \( 1 - 2.64T \)
good11 \( 1 + 18.1iT - 121T^{2} \)
13 \( 1 - 22.5T + 169T^{2} \)
17 \( 1 - 0.457iT - 289T^{2} \)
19 \( 1 - 30.1T + 361T^{2} \)
23 \( 1 + 12.3iT - 529T^{2} \)
29 \( 1 + 4.70iT - 841T^{2} \)
31 \( 1 - 45.2T + 961T^{2} \)
37 \( 1 + 32.9T + 1.36e3T^{2} \)
41 \( 1 + 22.9iT - 1.68e3T^{2} \)
43 \( 1 + 20.2T + 1.84e3T^{2} \)
47 \( 1 - 9.67iT - 2.20e3T^{2} \)
53 \( 1 - 5.97iT - 2.80e3T^{2} \)
59 \( 1 + 112. iT - 3.48e3T^{2} \)
61 \( 1 + 56.3T + 3.72e3T^{2} \)
67 \( 1 + 67.1T + 4.48e3T^{2} \)
71 \( 1 - 20.9iT - 5.04e3T^{2} \)
73 \( 1 - 97.7T + 5.32e3T^{2} \)
79 \( 1 - 41.4T + 6.24e3T^{2} \)
83 \( 1 - 121. iT - 6.88e3T^{2} \)
89 \( 1 + 26.8iT - 7.92e3T^{2} \)
97 \( 1 + 151.T + 9.40e3T^{2} \)
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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

−9.219545835697895836516837119749, −8.355894699366726234232258446937, −8.039088070209803533868692294730, −6.66662825381452027276394600716, −5.91416993437269436301634464800, −5.14300158844107857628155660777, −3.65532070371441034488671764523, −2.88176602052346747786818084582, −1.42590548909866247447317292652, −0.68723362475501690343582978294, 1.30812097605279049331012281590, 3.18109563631594288631688801289, 4.18088295347253028729682283897, 4.93672429659346924066617900390, 5.73285482569279568640077420327, 6.65759028780548529085508844425, 7.60646022989965382765964239361, 8.491508569186945994210750992001, 9.299905804370612214352981506614, 9.958921093666518094761859379000

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