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Properties

Label 2-1050-35.33-c1-0-12
Degree $2$
Conductor $1050$
Sign $0.992 + 0.124i$
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 + (−2.25 − 1.38i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (0.582 + 1.00i)11-s + (0.965 − 0.258i)12-s + (−1.92 + 1.92i)13-s + (0.756 − 2.53i)14-s + (0.500 − 0.866i)16-s + (0.00560 − 0.0209i)17-s + (−0.258 + 0.965i)18-s + (0.989 − 1.71i)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 + (−0.851 − 0.524i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.175 + 0.304i)11-s + (0.278 − 0.0747i)12-s + (−0.533 + 0.533i)13-s + (0.202 − 0.677i)14-s + (0.125 − 0.216i)16-s + (0.00136 − 0.00507i)17-s + (−0.0610 + 0.227i)18-s + (0.226 − 0.393i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.992 + 0.124i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.089417465\)
\(L(\frac12)\) \(\approx\) \(1.089417465\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
good11 \( 1 + (-0.582 - 1.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.92 - 1.92i)T - 13iT^{2} \)
17 \( 1 + (-0.00560 + 0.0209i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.989 + 1.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.93 + 1.85i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.60iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.74 + 10.2i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + (-7.87 - 7.87i)T + 43iT^{2} \)
47 \( 1 + (-3.94 + 1.05i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.757 + 2.82i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.34 + 9.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-14.0 - 3.76i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + (3.61 + 0.969i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.39 - 0.805i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.74 - 9.74i)T - 83iT^{2} \)
89 \( 1 + (1.80 - 3.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.265 + 0.265i)T + 97iT^{2} \)
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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

−9.667689804014754510715067692622, −9.253799186133240391955432331083, −7.987912803809880524482855497297, −7.10829841768320571630019917313, −6.65400570023527400157307907442, −5.75264259784403693071029088704, −4.72325585323037404596180206328, −3.95574248175459296859763546994, −2.58479522766484865212017904862, −0.62177675036484245482296148640, 1.06761920508976858836039714506, 2.75085053143453017733571143684, 3.46429383983755152165986813300, 4.77644850684409668762413358294, 5.49211623892789684735667037110, 6.38832662594366879987329883129, 7.28728387580494493194312107319, 8.587906287355837494303837342343, 9.259483785186888157746096155576, 10.12857788670809466480507478945

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