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Properties

Label 2-1050-21.17-c1-0-25
Degree $2$
Conductor $1050$
Sign $0.257 + 0.966i$
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  + (0.866 − 0.5i)2-s + (−1.66 − 0.464i)3-s + (0.499 − 0.866i)4-s + (−1.67 + 0.431i)6-s + (0.311 − 2.62i)7-s − 0.999i·8-s + (2.56 + 1.55i)9-s + (4.44 + 2.56i)11-s + (−1.23 + 1.21i)12-s + 5.00i·13-s + (−1.04 − 2.43i)14-s + (−0.5 − 0.866i)16-s + (1.87 − 3.25i)17-s + (2.99 + 0.0596i)18-s + (2.33 − 1.34i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.963 − 0.268i)3-s + (0.249 − 0.433i)4-s + (−0.684 + 0.176i)6-s + (0.117 − 0.993i)7-s − 0.353i·8-s + (0.855 + 0.517i)9-s + (1.34 + 0.774i)11-s + (−0.357 + 0.350i)12-s + 1.38i·13-s + (−0.279 − 0.649i)14-s + (−0.125 − 0.216i)16-s + (0.455 − 0.789i)17-s + (0.706 + 0.0140i)18-s + (0.534 − 0.308i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.863078589\)
\(L(\frac12)\) \(\approx\) \(1.863078589\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (1.66 + 0.464i)T \)
5 \( 1 \)
7 \( 1 + (-0.311 + 2.62i)T \)
good11 \( 1 + (-4.44 - 2.56i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.00iT - 13T^{2} \)
17 \( 1 + (-1.87 + 3.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.33 + 1.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.15 + 1.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.18iT - 29T^{2} \)
31 \( 1 + (4.13 + 2.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0262 - 0.0453i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.55T + 41T^{2} \)
43 \( 1 - 9.45T + 43T^{2} \)
47 \( 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.963 + 0.556i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.63 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.59 + 2.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.448 - 0.776i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (-6.04 - 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.37 - 14.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.37T + 83T^{2} \)
89 \( 1 + (2.67 + 4.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.633iT - 97T^{2} \)
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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

−9.731663857511428785470305955795, −9.397503558910276239536011200166, −7.69624549615169520426554846426, −6.87345296884780045780553670409, −6.53743261135943336134002349439, −5.25702518880862477731716255735, −4.41997669833752364387928584046, −3.83114859267330659977868469781, −2.02381579334035655225722656378, −0.947942220355894224274138118719, 1.29986501836474832243334007061, 3.15538159162765200227147289050, 3.93059058554902799507823582598, 5.32082121577161405300446118541, 5.60011712726144714915418401129, 6.41373068344773044701677199001, 7.35450920914593050730829860528, 8.465694690042744018983591092290, 9.173563989788340053516426423671, 10.26662461254260323975855024559

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