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Properties

Label 2-1050-35.4-c1-0-20
Degree $2$
Conductor $1050$
Sign $0.669 + 0.742i$
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 + (1.73 − 2i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 − 0.499i)12-s + i·13-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−1.5 − 2.59i)19-s + (2.5 − 0.866i)21-s − 0.999i·22-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 + (0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s + (0.249 − 0.144i)12-s + 0.277i·13-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.204 + 0.117i)18-s + (−0.344 − 0.596i)19-s + (0.545 − 0.188i)21-s − 0.213i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.669 + 0.742i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.939677323\)
\(L(\frac12)\) \(\approx\) \(2.939677323\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 2i)T \)
good11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.52 - 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (-4.33 + 2.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.52 - 5.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−10.09291115512357695302270059906, −8.883949541190483146756096409864, −8.330959702442013915776145955876, −7.13836915180654795798899271786, −6.51719435760704275130596534428, −5.06810510057466644561692028371, −4.55530121253917408076947501295, −3.54871004938197672819202630355, −2.54925699733949294482816926016, −1.18596051240309613697934638312, 1.65171150572162735313917037641, 2.76287387021183708056156735574, 3.78484761449542754336819170611, 4.95534410521548148420021826264, 5.61029690597784776309569791196, 6.75345232711337025105040038236, 7.39610385892864508068016236675, 8.523320025912606676675573087333, 8.736163772826644409564662605933, 10.02138584811831310278669236716

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