Properties

Label 2-1050-105.104-c1-0-2
Degree $2$
Conductor $1050$
Sign $-0.826 + 0.562i$
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  − 2-s + (−0.618 + 1.61i)3-s + 4-s + (0.618 − 1.61i)6-s + (−0.381 + 2.61i)7-s − 8-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (−0.618 + 1.61i)12-s − 1.23·13-s + (0.381 − 2.61i)14-s + 16-s + 5.23i·17-s + (2.23 + 2.00i)18-s − 8.47i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.356 + 0.934i)3-s + 0.5·4-s + (0.252 − 0.660i)6-s + (−0.144 + 0.989i)7-s − 0.353·8-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (−0.178 + 0.467i)12-s − 0.342·13-s + (0.102 − 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (0.527 + 0.471i)18-s − 1.94i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3665324487\)
\(L(\frac12)\) \(\approx\) \(0.3665324487\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.618 - 1.61i)T \)
5 \( 1 \)
7 \( 1 + (0.381 - 2.61i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 5.23iT - 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 - 0.763iT - 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 - 4.94iT - 43T^{2} \)
47 \( 1 + 6.47iT - 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 + 0.763T + 97T^{2} \)
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

−10.31569756824204548140722082373, −9.473609144747179844357987109218, −9.047858108428732770761846180308, −8.173584138410875123448474901653, −7.01444861664052879383588329319, −6.22329016381309338527679419472, −5.21247156413096119042424969695, −4.42223766058009006724846012396, −3.08077015514314438661913254244, −1.98216401048686135007418113949, 0.22166513089523079360118300480, 1.35128652297044258297416146715, 2.72990236981171024331512826361, 3.90683196790939238006544323023, 5.45245079835018980185444640790, 6.19544743646273025421920620582, 7.01770567412587355292641457327, 7.890310624605009689549810657152, 8.240667803116094571609073814558, 9.489693209870684712427752314542

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