Properties

Label 2-429-429.428-c1-0-7
Degree $2$
Conductor $429$
Sign $-0.310 + 0.950i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
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  + 2.13i·2-s + (0.780 + 1.54i)3-s − 2.56·4-s − 3.09·5-s + (−3.30 + 1.66i)6-s − 1.19i·8-s + (−1.78 + 2.41i)9-s − 6.60i·10-s + (−3.09 − 1.19i)11-s + (−1.99 − 3.96i)12-s + (3.30 + 1.44i)13-s + (−2.41 − 4.78i)15-s − 2.56·16-s + 5.73·17-s + (−5.15 − 3.80i)18-s − 2.89·19-s + ⋯
L(s)  = 1  + 1.51i·2-s + (0.450 + 0.892i)3-s − 1.28·4-s − 1.38·5-s + (−1.34 + 0.680i)6-s − 0.424i·8-s + (−0.593 + 0.804i)9-s − 2.08i·10-s + (−0.932 − 0.361i)11-s + (−0.577 − 1.14i)12-s + (0.915 + 0.401i)13-s + (−0.623 − 1.23i)15-s − 0.640·16-s + 1.39·17-s + (−1.21 − 0.896i)18-s − 0.664·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.310 + 0.950i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.310 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474237 - 0.653699i\)
\(L(\frac12)\) \(\approx\) \(0.474237 - 0.653699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.780 - 1.54i)T \)
11 \( 1 + (3.09 + 1.19i)T \)
13 \( 1 + (-3.30 - 1.44i)T \)
good2 \( 1 - 2.13iT - 2T^{2} \)
5 \( 1 + 3.09T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + 3.09iT - 23T^{2} \)
29 \( 1 + 5.73T + 29T^{2} \)
31 \( 1 - 2.68iT - 31T^{2} \)
37 \( 1 - 9.56iT - 37T^{2} \)
41 \( 1 - 5.20iT - 41T^{2} \)
43 \( 1 - 6.60iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6.18iT - 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 2.89iT - 61T^{2} \)
67 \( 1 + 2.68iT - 67T^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + 0.813iT - 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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

−11.55216076304689567754569074374, −10.89266580566122674707068202221, −9.751278263273940611287631286288, −8.451706859121024364083185900205, −8.244647559986197946852615292888, −7.42633250994198059185732994773, −6.17002700727369813208821751453, −5.11205228779103569729711773972, −4.21415053857489397887566040790, −3.16255744722960631899498831524, 0.48493225825512684839579867221, 2.02540687030513881673439672244, 3.34835642622391660648918886263, 3.88005418022550546832087267628, 5.56312696271301494914177328568, 7.18487833087758450321835448862, 7.88804165820202014177221877197, 8.696213767758395636338859060864, 9.826663003775665808682185677701, 10.87685444629182254165377735690

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