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Properties

Label 2-429-13.3-c1-0-7
Degree $2$
Conductor $429$
Sign $-0.298 + 0.954i$
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  + (−1.19 − 2.07i)2-s + (−0.5 − 0.866i)3-s + (−1.86 + 3.23i)4-s + 3.82·5-s + (−1.19 + 2.07i)6-s + (−2.02 + 3.51i)7-s + 4.16·8-s + (−0.499 + 0.866i)9-s + (−4.57 − 7.93i)10-s + (0.5 + 0.866i)11-s + 3.73·12-s + (1.95 − 3.03i)13-s + 9.72·14-s + (−1.91 − 3.31i)15-s + (−1.25 − 2.16i)16-s + (2.90 − 5.03i)17-s + ⋯
L(s)  = 1  + (−0.846 − 1.46i)2-s + (−0.288 − 0.499i)3-s + (−0.934 + 1.61i)4-s + 1.70·5-s + (−0.488 + 0.846i)6-s + (−0.767 + 1.32i)7-s + 1.47·8-s + (−0.166 + 0.288i)9-s + (−1.44 − 2.50i)10-s + (0.150 + 0.261i)11-s + 1.07·12-s + (0.541 − 0.841i)13-s + 2.59·14-s + (−0.493 − 0.854i)15-s + (−0.312 − 0.541i)16-s + (0.704 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591517 - 0.805044i\)
\(L(\frac12)\) \(\approx\) \(0.591517 - 0.805044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-1.95 + 3.03i)T \)
good2 \( 1 + (1.19 + 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.82T + 5T^{2} \)
7 \( 1 + (2.02 - 3.51i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-2.90 + 5.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.85 + 4.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 4.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.68 + 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.05T + 31T^{2} \)
37 \( 1 + (0.752 + 1.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.74 - 4.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.55 + 2.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.97T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + (2.01 - 3.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.27 - 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.82 - 3.16i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.00632T + 73T^{2} \)
79 \( 1 + 3.36T + 79T^{2} \)
83 \( 1 - 8.92T + 83T^{2} \)
89 \( 1 + (-6.51 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.68 + 16.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.83956416956443262467586544342, −9.824606909652447646139663053686, −9.381838139752993704017518830803, −8.794554167033825613748064966007, −7.30914827507166983878144801078, −5.95563983900327400538486773700, −5.35022369343814898682105944014, −2.99115998847986926577738239946, −2.46225539904750411952224019292, −1.12971012861988001999635617692, 1.25844256579875004196599154837, 3.70593729777373703973332109515, 5.20539163924865697430630551834, 6.21947336117908363961366600619, 6.45681176247407412148982902279, 7.63736224084560054052208891960, 8.899233532404810817973037998223, 9.489704854841037786011164134698, 10.30355299973590222250337633774, 10.63887524742869266248769034567

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