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Properties

Label 2-429-429.428-c1-0-29
Degree $2$
Conductor $429$
Sign $0.488 - 0.872i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.488 - 0.872i$
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: \((2,\ 429,\ (\ :1/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65956 + 0.972316i\)
\(L(\frac12)\) \(\approx\) \(1.65956 + 0.972316i\)
\(L(\frac{3}{2})\) 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 \( 1 + 14.9iT - 97T^{2} \)
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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

−11.51871178026258844215749957008, −9.812692044097746668296962550151, −9.363519280828004218339701004034, −8.285714201178082104061198286068, −7.48485664628295444739877186795, −6.65538695416473339428363708258, −5.86246692042079166420130204746, −5.19830659625384911592670239353, −3.16984367289827124159220577192, −1.58670731477108921553325839032, 1.68418432584048907989163730209, 2.65750893984742658608490561371, 3.77703072060143718017885237128, 4.85508074008550229120603219370, 5.93082790701693356910363163042, 7.46973272901721722021567253713, 9.125810396174553581903397603975, 9.474057885730831224608586952979, 10.02830496969328142260994290467, 10.81164677686699517998408952595

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