Properties

Label 2-1050-15.14-c2-0-32
Degree $2$
Conductor $1050$
Sign $0.983 + 0.180i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−0.836 − 2.88i)3-s + 2.00·4-s + (−1.18 − 4.07i)6-s + 2.64i·7-s + 2.82·8-s + (−7.60 + 4.81i)9-s + 13.1i·11-s + (−1.67 − 5.76i)12-s − 6.59i·13-s + 3.74i·14-s + 4.00·16-s + 21.7·17-s + (−10.7 + 6.81i)18-s + 5.34·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.278 − 0.960i)3-s + 0.500·4-s + (−0.197 − 0.679i)6-s + 0.377i·7-s + 0.353·8-s + (−0.844 + 0.535i)9-s + 1.19i·11-s + (−0.139 − 0.480i)12-s − 0.507i·13-s + 0.267i·14-s + 0.250·16-s + 1.27·17-s + (−0.597 + 0.378i)18-s + 0.281·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.983 + 0.180i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.775276366\)
\(L(\frac12)\) \(\approx\) \(2.775276366\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (0.836 + 2.88i)T \)
5 \( 1 \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 13.1iT - 121T^{2} \)
13 \( 1 + 6.59iT - 169T^{2} \)
17 \( 1 - 21.7T + 289T^{2} \)
19 \( 1 - 5.34T + 361T^{2} \)
23 \( 1 - 0.0360T + 529T^{2} \)
29 \( 1 - 0.0748iT - 841T^{2} \)
31 \( 1 - 36.8T + 961T^{2} \)
37 \( 1 - 71.7iT - 1.36e3T^{2} \)
41 \( 1 - 3.25iT - 1.68e3T^{2} \)
43 \( 1 + 49.0iT - 1.84e3T^{2} \)
47 \( 1 - 58.2T + 2.20e3T^{2} \)
53 \( 1 - 93.5T + 2.80e3T^{2} \)
59 \( 1 - 49.6iT - 3.48e3T^{2} \)
61 \( 1 + 43.0T + 3.72e3T^{2} \)
67 \( 1 + 68.3iT - 4.48e3T^{2} \)
71 \( 1 + 87.2iT - 5.04e3T^{2} \)
73 \( 1 + 1.12iT - 5.32e3T^{2} \)
79 \( 1 + 70.2T + 6.24e3T^{2} \)
83 \( 1 + 0.397T + 6.88e3T^{2} \)
89 \( 1 - 1.85iT - 7.92e3T^{2} \)
97 \( 1 - 34.4iT - 9.40e3T^{2} \)
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

−9.923635188028292012964947544275, −8.649711348728730677757193472871, −7.74095341068959572856884679385, −7.16207965155860156737753609196, −6.21717261922811326992840124833, −5.45677372647016556545593900904, −4.65478575336890181380268058668, −3.23343863639041731945870116612, −2.28678126705051002878058746482, −1.10869842136843914732899958624, 0.856522310401162140117166670492, 2.76528158053170355771261137564, 3.65904310068311762307965123276, 4.37272039188628364581542135084, 5.50825190817615953954695472427, 5.94197015915981993280367226286, 7.07334767730419295618593357665, 8.119198355228477103034033021135, 9.014444179097280269444407160091, 9.939449021448767954189836669079

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