Properties

Label 2-1050-35.12-c1-0-13
Degree $2$
Conductor $1050$
Sign $0.964 + 0.265i$
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 + (1.48 − 2.19i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.366 + 0.633i)11-s + (0.258 + 0.965i)12-s + (1.03 + 1.03i)13-s + (0.866 − 2.49i)14-s + (0.500 − 0.866i)16-s + (2.19 + 0.586i)17-s + (−0.965 − 0.258i)18-s + (2.09 − 3.63i)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 + (0.560 − 0.827i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.110 + 0.191i)11-s + (0.0747 + 0.278i)12-s + (0.287 + 0.287i)13-s + (0.231 − 0.668i)14-s + (0.125 − 0.216i)16-s + (0.531 + 0.142i)17-s + (−0.227 − 0.0610i)18-s + (0.481 − 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.964 + 0.265i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.554675400\)
\(L(\frac12)\) \(\approx\) \(2.554675400\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.19i)T \)
good11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.03 - 1.03i)T + 13iT^{2} \)
17 \( 1 + (-2.19 - 0.586i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.09 + 3.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.965 - 3.60i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.19iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.57 - 1.22i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.46iT - 41T^{2} \)
43 \( 1 + (-0.138 + 0.138i)T - 43iT^{2} \)
47 \( 1 + (-2.84 - 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.08 - 1.36i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.267 - 0.464i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.5 - 6.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.03 - 3.86i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-2.07 + 7.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.03 + 4.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \)
89 \( 1 + (-3.23 + 5.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.94 - 4.94i)T - 97iT^{2} \)
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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

−10.03825725014294994299983875330, −9.286028686327499779016385430511, −8.098494442131497277595186932762, −7.31833072452070631624445593698, −6.33588192757601895914680961138, −5.39166594733083795845689546622, −4.48828422523992951419678896759, −3.88132804079057508032494925476, −2.69027606353788717992481458203, −1.14826936970928191543563989337, 1.39541726065233977853179457013, 2.63714509130197355885867135157, 3.65524810801166340902996256923, 5.05307632367368406540478033736, 5.53553937437337414648678603424, 6.49278700932348584035436183642, 7.30461365376155432345992612081, 8.305918405280512698993684257224, 8.759893454817547675869096065819, 10.13110702331847225651758680954

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