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Properties

Label 2-1050-35.3-c1-0-18
Degree $2$
Conductor $1050$
Sign $0.909 + 0.416i$
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  + (0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (2.63 − 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.17 − 2.04i)11-s + (0.258 − 0.965i)12-s + (−0.0968 + 0.0968i)13-s + (2.61 + 0.431i)14-s + (0.500 + 0.866i)16-s + (3.11 − 0.833i)17-s + (−0.965 + 0.258i)18-s + (−0.434 − 0.752i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (0.995 − 0.0978i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.355 − 0.615i)11-s + (0.0747 − 0.278i)12-s + (−0.0268 + 0.0268i)13-s + (0.697 + 0.115i)14-s + (0.125 + 0.216i)16-s + (0.754 − 0.202i)17-s + (−0.227 + 0.0610i)18-s + (−0.0996 − 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.649810616\)
\(L(\frac12)\) \(\approx\) \(2.649810616\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.63 + 0.258i)T \)
good11 \( 1 + (-1.17 + 2.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0968 - 0.0968i)T - 13iT^{2} \)
17 \( 1 + (-3.11 + 0.833i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.434 + 0.752i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.833 - 3.11i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.08iT - 29T^{2} \)
31 \( 1 + (0.747 + 0.431i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.51 - 2.54i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.86iT - 41T^{2} \)
43 \( 1 + (-2.57 - 2.57i)T + 43iT^{2} \)
47 \( 1 + (-0.223 + 0.833i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.07 - 2.16i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.35 + 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.44 - 4.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 12.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + (4.20 + 15.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.02 - 2.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.13 - 7.13i)T - 83iT^{2} \)
89 \( 1 + (4.98 + 8.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \)
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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

−9.927673329867766761896832208509, −8.824487513536828351607234639957, −7.906881877006796086585739890636, −7.43441432822059744863391625857, −6.30513991186625009997721914403, −5.62646678368119320804439618350, −4.70947562530188867873730878826, −3.68625481891426199239252779590, −2.45169076014316144687351620901, −1.18490471125347180775511616800, 1.45305104885622600749290826195, 2.73266986618539166654942936528, 3.97972261891600432274554994539, 4.65799353030933387525403268764, 5.47279731803433249210202544336, 6.36335678553221728155385228183, 7.46337477731742187121379879684, 8.268851965039353195859662112232, 9.306046006880526895785787988889, 10.11241744816861088708714019651

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