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Properties

Label 2-1050-35.12-c1-0-22
Degree $2$
Conductor $1050$
Sign $-0.810 + 0.585i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (−2.63 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.54 − 4.40i)11-s + (0.258 + 0.965i)12-s + (−2.02 − 2.02i)13-s + (−2.47 + 0.931i)14-s + (0.500 − 0.866i)16-s + (−6.71 − 1.79i)17-s + (−0.965 − 0.258i)18-s + (−1.79 + 3.11i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (−0.995 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.767 − 1.32i)11-s + (0.0747 + 0.278i)12-s + (−0.561 − 0.561i)13-s + (−0.661 + 0.248i)14-s + (0.125 − 0.216i)16-s + (−1.62 − 0.436i)17-s + (−0.227 − 0.0610i)18-s + (−0.412 + 0.714i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6109033030\)
\(L(\frac12)\) \(\approx\) \(0.6109033030\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (2.63 - 0.258i)T \)
good11 \( 1 + (2.54 + 4.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.02 + 2.02i)T + 13iT^{2} \)
17 \( 1 + (6.71 + 1.79i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.79 - 6.71i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.81iT - 29T^{2} \)
31 \( 1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.50 - 2.01i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.13iT - 41T^{2} \)
43 \( 1 + (-7.84 + 7.84i)T - 43iT^{2} \)
47 \( 1 + (0.482 + 1.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.879 + 0.235i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.762 - 2.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.86T + 71T^{2} \)
73 \( 1 + (3.50 - 13.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.4 + 6.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.99 - 1.99i)T + 83iT^{2} \)
89 \( 1 + (-3.82 + 6.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.4 - 13.4i)T - 97iT^{2} \)
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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

−9.757671717442153072779353695801, −8.903426433593636262988048352235, −7.919256329831398668754221886465, −6.80558783170711416357606979962, −5.91396775987100844168142813684, −5.34172520465401613057796126660, −4.18690398451688921884357983339, −3.28455976861678910797693473528, −2.46255744477638733448317946906, −0.19830459316095912694763586583, 2.07532545694154417790593529514, 2.84762141282124378179167312787, 4.34547453714575308475911805643, 4.90940716856902714528228479221, 6.19502841959877572184255974925, 6.93712803720138176796043001729, 7.23139738323793772986083891315, 8.593315029224634634158117550983, 9.322643227650508620262455919762, 10.51097057107083507869171326873

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