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Properties

Label 2-1050-35.17-c1-0-5
Degree $2$
Conductor $1050$
Sign $0.918 - 0.396i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.918 - 0.396i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.918 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.298382477\)
\(L(\frac12)\) \(\approx\) \(1.298382477\)
\(L(\frac{3}{2})\) 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

−10.06939213828687470266381937229, −9.501121268038728879343971901264, −8.213168602200006198918627695596, −7.68883058544269674629220244307, −6.41545641237889945051879931760, −5.23613076373182959269548213171, −4.96810371824138838977337374757, −3.75925895690866733354443960617, −2.48639373435427732234983282193, −1.33514249443572818487538618343, 0.65420726522704859423327664357, 2.40322388368395202727345514713, 3.92318938810404865895228913064, 4.91522994108371040088002403442, 5.43718852569204951491051424583, 6.55236520366059060245625238186, 7.29505467952614570324356775157, 8.008538690846452360657701279684, 8.859107758064635438145982092933, 9.848625755975661640287634770104

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