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Properties

Label 2-1050-35.17-c1-0-9
Degree $2$
Conductor $1050$
Sign $0.466 - 0.884i$
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 + (−0.153 + 2.64i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (2.27 − 3.94i)11-s + (−0.965 − 0.258i)12-s + (−1.77 − 1.77i)13-s + (−2.51 − 0.831i)14-s + (0.500 + 0.866i)16-s + (1.06 + 3.98i)17-s + (0.258 + 0.965i)18-s + (1.88 + 3.27i)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 + (−0.0579 + 0.998i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.686 − 1.18i)11-s + (−0.278 − 0.0747i)12-s + (−0.493 − 0.493i)13-s + (−0.671 − 0.222i)14-s + (0.125 + 0.216i)16-s + (0.258 + 0.966i)17-s + (0.0610 + 0.227i)18-s + (0.433 + 0.750i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.466 - 0.884i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.776944326\)
\(L(\frac12)\) \(\approx\) \(1.776944326\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.153 - 2.64i)T \)
good11 \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.77 + 1.77i)T + 13iT^{2} \)
17 \( 1 + (-1.06 - 3.98i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.88 - 3.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.77 - 2.08i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.55iT - 29T^{2} \)
31 \( 1 + (-3.37 - 1.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.95 + 11.0i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (0.367 - 0.367i)T - 43iT^{2} \)
47 \( 1 + (4.87 + 1.30i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.18 - 8.14i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.221 - 0.383i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.09 + 4.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.99 + 2.41i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (-4.20 + 1.12i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.08 - 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.21 - 3.21i)T + 83iT^{2} \)
89 \( 1 + (-3.02 - 5.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.462 - 0.462i)T - 97iT^{2} \)
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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

−9.688567404403392754483877021477, −9.077303445347422433178010577399, −8.366649282747447337613663100694, −7.76414541073451564978065964333, −6.64995538329763456519551324166, −5.88835969552073737191540110745, −5.13215772611014221324340941287, −3.72596475654276208326014165625, −2.83984858055366638999023671211, −1.25739154299805245458832659784, 0.992007257959244881940142298376, 2.33947660258841269857837968761, 3.35819584876023789550223608652, 4.44695248134224338949862902598, 4.93266335728800596666513248192, 6.93770091038144437533882115397, 7.10074662070505776405940215282, 8.251851536971212081048366170879, 9.282787491532129741107876737232, 9.698986625499521349912935626856

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