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Properties

Label 2-1050-7.4-c1-0-9
Degree $2$
Conductor $1050$
Sign $0.968 + 0.250i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + (−2.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + (−0.499 − 0.866i)12-s + 13-s + (2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.144 − 0.249i)12-s + 0.277·13-s + (0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.117 + 0.204i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.020343263\)
\(L(\frac12)\) \(\approx\) \(1.020343263\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−9.878539762188459890328668583093, −9.070687482885223001763757922645, −8.697693142118292960419602707999, −7.45260312017594007889447383619, −6.29677017624464986440904984779, −5.74316879069052146843386764901, −4.34333851147284749707892233547, −3.53535842091399441853401678485, −2.60569168306205982818148576472, −0.838615799817317477900438722436, 0.835910673877510865392407794410, 2.35720735954868266836583136696, 3.85616744471844998033939457276, 4.85646338391146831939142167997, 6.00990740487378164892421728331, 6.70433383303470846135701217114, 7.22686566641299084553236629991, 8.172840002616628145709527841324, 9.186146840794650617902148599649, 9.762356199163517965747611310847

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