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Properties

Label 2-1050-35.4-c1-0-13
Degree $2$
Conductor $1050$
Sign $0.997 - 0.0667i$
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 + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (2.59 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s − 5i·13-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (5.19 + 3i)17-s + (−0.866 − 0.499i)18-s + (−3.5 − 6.06i)19-s + (2.5 + 0.866i)21-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (0.981 − 0.188i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.249 − 0.144i)12-s − 1.38i·13-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (1.26 + 0.727i)17-s + (−0.204 − 0.117i)18-s + (−0.802 − 1.39i)19-s + (0.545 + 0.188i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.997 - 0.0667i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.997 - 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.626018491\)
\(L(\frac12)\) \(\approx\) \(1.626018491\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.5i)T \)
good11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-4.33 - 2.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−9.944458665609951579461867895278, −8.781307816395753312569489965807, −8.435212595220303902025231607635, −7.58673709179306951530582031722, −6.84221814946536688479406875908, −5.50118048357666757175082831655, −4.90980299174871035409226294637, −3.58827250743380209070156878801, −2.41988613153135665465514825266, −1.00137273142647551527303110101, 1.34414846601676598348099111557, 2.18638809054324946269286527763, 3.47032541989576989459121377361, 4.48290495425953560113770976415, 5.67687299346993898968187924592, 6.80690376075500429784721705736, 7.74073239548669749574967608567, 8.143853125775977824519808080170, 9.259323646695044196797837347002, 9.574900546154250697648729073542

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