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Properties

Label 2-1050-21.5-c1-0-33
Degree $2$
Conductor $1050$
Sign $-0.995 + 0.0978i$
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 + (−1.20 − 1.24i)3-s + (0.499 + 0.866i)4-s + (0.421 + 1.68i)6-s + (−0.0551 + 2.64i)7-s − 0.999i·8-s + (−0.0969 + 2.99i)9-s + (0.167 − 0.0969i)11-s + (0.475 − 1.66i)12-s + 1.54i·13-s + (1.37 − 2.26i)14-s + (−0.5 + 0.866i)16-s + (0.264 + 0.458i)17-s + (1.58 − 2.54i)18-s + (−5.53 − 3.19i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.695 − 0.718i)3-s + (0.249 + 0.433i)4-s + (0.171 + 0.685i)6-s + (−0.0208 + 0.999i)7-s − 0.353i·8-s + (−0.0323 + 0.999i)9-s + (0.0506 − 0.0292i)11-s + (0.137 − 0.480i)12-s + 0.429i·13-s + (0.366 − 0.604i)14-s + (−0.125 + 0.216i)16-s + (0.0642 + 0.111i)17-s + (0.373 − 0.600i)18-s + (−1.26 − 0.732i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.995 + 0.0978i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.995 + 0.0978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2601025000\)
\(L(\frac12)\) \(\approx\) \(0.2601025000\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 1.24i)T \)
5 \( 1 \)
7 \( 1 + (0.0551 - 2.64i)T \)
good11 \( 1 + (-0.167 + 0.0969i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.54iT - 13T^{2} \)
17 \( 1 + (-0.264 - 0.458i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.53 + 3.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.87iT - 29T^{2} \)
31 \( 1 + (-8.02 + 4.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.881 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.91T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-4.90 + 8.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0562 + 0.0324i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.01 + 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 - 2.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (-7.00 + 4.04i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.38 - 5.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.11T + 83T^{2} \)
89 \( 1 + (8.18 - 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.24iT - 97T^{2} \)
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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

−9.615544323527825352814656706896, −8.374849177851875018475280266968, −8.245770331407972898472998373770, −6.79857303381980389391213582434, −6.38673562452844221772589532267, −5.33084892377096378469487312265, −4.23552930393939331004867879237, −2.59550765112063489597159751110, −1.85589738215735874360451032660, −0.16167693481176023676069774846, 1.35655226603696977126356630154, 3.28203448761687782915505047397, 4.31456052455844271351191207435, 5.17612158272932917403045407108, 6.26178911315311325539892949043, 6.80922780162952341445405899509, 7.919574585213317622425050162607, 8.648249634327994799054944086756, 9.720480647519535318956054956609, 10.34755244421774487557484410033

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