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Properties

Label 2-1050-35.17-c1-0-22
Degree $2$
Conductor $1050$
Sign $-0.779 + 0.626i$
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 + (−0.258 − 2.63i)7-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.17 − 2.04i)11-s + (−0.965 − 0.258i)12-s + (−0.0968 − 0.0968i)13-s + (−2.61 − 0.431i)14-s + (0.500 + 0.866i)16-s + (−0.833 − 3.11i)17-s + (−0.258 − 0.965i)18-s + (0.434 + 0.752i)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 + (−0.0978 − 0.995i)7-s + (−0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.355 − 0.615i)11-s + (−0.278 − 0.0747i)12-s + (−0.0268 − 0.0268i)13-s + (−0.697 − 0.115i)14-s + (0.125 + 0.216i)16-s + (−0.202 − 0.754i)17-s + (−0.0610 − 0.227i)18-s + (0.0996 + 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.779 + 0.626i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.790167091\)
\(L(\frac12)\) \(\approx\) \(1.790167091\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 2.63i)T \)
good11 \( 1 + (-1.17 + 2.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0968 + 0.0968i)T + 13iT^{2} \)
17 \( 1 + (0.833 + 3.11i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.434 - 0.752i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.11 + 0.833i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.08iT - 29T^{2} \)
31 \( 1 + (0.747 + 0.431i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.54 + 9.51i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.86iT - 41T^{2} \)
43 \( 1 + (2.57 - 2.57i)T - 43iT^{2} \)
47 \( 1 + (0.833 + 0.223i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.16 + 8.07i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (-12.2 + 3.28i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + (-15.7 + 4.20i)T + (63.2 - 36.5i)T^{2} \)
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

−9.590631511414178100508353177388, −8.958629028266420430044540482580, −7.976205466646822335847923833218, −7.17997602027238441930317007940, −6.21622836390167197542401897206, −5.02084447228099152352327552129, −3.96783856127016290046799256579, −3.31847183566045475749135137041, −2.07425643145534539418193563551, −0.71451695463501772149758708691, 1.88981064512488575826402419498, 3.08431675276224536725559172146, 4.18961734539246922546066094995, 5.05522896894772150369808116898, 6.11768433670678080479500945669, 6.76043161850079502946839834122, 7.993590261910614890218776367688, 8.347402323336072824230336505247, 9.474331628840145203189962041876, 9.744174880574240365823791469123

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