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Properties

Label 2-1050-35.17-c1-0-4
Degree $2$
Conductor $1050$
Sign $-0.204 - 0.978i$
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 + (−2.46 − 0.965i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (−1.55 + 2.69i)11-s + (−0.965 − 0.258i)12-s + (2.30 + 2.30i)13-s + (1.57 − 2.12i)14-s + (0.500 + 0.866i)16-s + (−0.295 − 1.10i)17-s + (0.258 + 0.965i)18-s + (2.99 + 5.18i)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 + (−0.930 − 0.365i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.469 + 0.813i)11-s + (−0.278 − 0.0747i)12-s + (0.639 + 0.639i)13-s + (0.419 − 0.569i)14-s + (0.125 + 0.216i)16-s + (−0.0716 − 0.267i)17-s + (0.0610 + 0.227i)18-s + (0.687 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.346774887\)
\(L(\frac12)\) \(\approx\) \(1.346774887\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.965i)T \)
good11 \( 1 + (1.55 - 2.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.30 - 2.30i)T + 13iT^{2} \)
17 \( 1 + (0.295 + 1.10i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.99 - 5.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.10 - 0.295i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.39iT - 29T^{2} \)
31 \( 1 + (-3.68 - 2.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.48 - 5.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (-6.44 + 6.44i)T - 43iT^{2} \)
47 \( 1 + (4.11 + 1.10i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.746 - 2.78i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.117 - 0.203i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.38 + 4.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.4 - 3.60i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.74T + 71T^{2} \)
73 \( 1 + (1.09 - 0.293i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.76 + 5.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.06 + 8.06i)T + 83iT^{2} \)
89 \( 1 + (-9.18 - 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.740 + 0.740i)T - 97iT^{2} \)
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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

−9.915893560775941492312450444121, −9.298744478630267524000372964016, −8.434006471687055552689373823896, −7.57642697400233613687425286782, −6.89258890315629592549659637447, −6.16086015428611985016112868657, −5.00027244179180863368611502687, −3.95183956782990973698701087478, −2.98128891579702322603457716987, −1.40786911480360551169720011895, 0.64792555124381414873770926381, 2.46370388198698365035652264668, 3.13656515062803757791632655856, 4.02810435035758135256569031697, 5.33324179152281515486554085080, 6.19639447364303581600895334821, 7.38336595706627201863716643895, 8.291300390514711939344537473218, 8.985169142416498522258181491194, 9.621806797728911709081369090992

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