Properties

Label 2-1050-35.33-c1-0-23
Degree $2$
Conductor $1050$
Sign $-0.354 + 0.935i$
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  + (−0.258 − 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s i·6-s + (1.87 − 1.86i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−2.74 − 4.75i)11-s + (−0.965 + 0.258i)12-s + (2.41 − 2.41i)13-s + (−2.28 − 1.33i)14-s + (0.500 − 0.866i)16-s + (0.548 − 2.04i)17-s + (0.258 − 0.965i)18-s + (−3.49 + 6.05i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s − 0.408i·6-s + (0.709 − 0.704i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.827 − 1.43i)11-s + (−0.278 + 0.0747i)12-s + (0.670 − 0.670i)13-s + (−0.611 − 0.355i)14-s + (0.125 − 0.216i)16-s + (0.132 − 0.496i)17-s + (0.0610 − 0.227i)18-s + (−0.802 + 1.38i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.354 + 0.935i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.354 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636014554\)
\(L(\frac12)\) \(\approx\) \(1.636014554\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.86i)T \)
good11 \( 1 + (2.74 + 4.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.41 + 2.41i)T - 13iT^{2} \)
17 \( 1 + (-0.548 + 2.04i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.49 - 6.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.69 - 0.454i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.684iT - 29T^{2} \)
31 \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.53 + 9.46i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.50iT - 41T^{2} \)
43 \( 1 + (-1.95 - 1.95i)T + 43iT^{2} \)
47 \( 1 + (-3.40 + 0.912i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.43 + 9.08i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.08 + 8.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 0.585i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.61 - 2.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (4.70 + 1.26i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.21 - 4.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.05 + 4.05i)T - 83iT^{2} \)
89 \( 1 + (3.59 - 6.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 - 13.1i)T + 97iT^{2} \)
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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

−9.849719029293657707695258363331, −8.688013876108313477799938981313, −8.138810817720527575805198801672, −7.64085560446783619558228500319, −6.12778735796122858546217633783, −5.20908396296449465466107555234, −4.00689502855195183925520303973, −3.35155261758893187765760799189, −2.15864699441625006331241913426, −0.75862732957774951431961051024, 1.67198831943769035359969067039, 2.67647926740971132344528561780, 4.33535885942409932552279918947, 4.87769480278847466415168558249, 6.06348397093690543252253557474, 6.95434831862798213311321706367, 7.69991131791121597237581736625, 8.586543165023876459121708556801, 8.973935209462239090427895678418, 10.03134686691640413841125592519

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