Properties

Label 2-1050-35.27-c1-0-16
Degree $2$
Conductor $1050$
Sign $0.835 - 0.550i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s − 1.00i·6-s + (2.63 + 0.189i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + 1.46·11-s + (0.707 − 0.707i)12-s + (−0.189 − 0.189i)13-s + (1.73 + 2i)14-s − 1.00·16-s + (3.53 − 3.53i)17-s + (−0.707 + 0.707i)18-s − 0.535·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s − 0.408i·6-s + (0.997 + 0.0716i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + 0.441·11-s + (0.204 − 0.204i)12-s + (−0.0525 − 0.0525i)13-s + (0.462 + 0.534i)14-s − 0.250·16-s + (0.857 − 0.857i)17-s + (−0.166 + 0.166i)18-s − 0.122·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.835 - 0.550i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.118164501\)
\(L(\frac12)\) \(\approx\) \(2.118164501\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.63 - 0.189i)T \)
good11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (0.189 + 0.189i)T + 13iT^{2} \)
17 \( 1 + (-3.53 + 3.53i)T - 17iT^{2} \)
19 \( 1 + 0.535T + 19T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - 23iT^{2} \)
29 \( 1 - 4.26iT - 29T^{2} \)
31 \( 1 - 5.92iT - 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 - 2.26iT - 41T^{2} \)
43 \( 1 + (-0.189 + 0.189i)T - 43iT^{2} \)
47 \( 1 + (-8.76 + 8.76i)T - 47iT^{2} \)
53 \( 1 + (-7.39 + 7.39i)T - 53iT^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 + (-10.1 - 10.1i)T + 67iT^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + (5.93 + 5.93i)T + 73iT^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (-4.57 - 4.57i)T + 83iT^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (11.9 - 11.9i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06952648392244926844213716195, −8.897124855032783118398553019268, −8.191330934761906190334347371830, −7.30198693084520113751473445694, −6.71311323733180459904580493660, −5.53373825353571662225153105446, −5.05109030093069686625924136116, −3.98437528678074402028123158057, −2.67010097597050345555287526034, −1.22844745592754389664772883559, 1.11489601145437531916503960706, 2.40372060414962530840432400730, 3.86927979246310166184437756059, 4.38447788014685018711398620004, 5.51128376666108550222390494971, 6.05855787800239047721104276969, 7.33581264436849187758118194806, 8.205386945688707910673134744013, 9.233711629982111129957045440179, 9.995965231853659736676709913371

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