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Properties

Label 2-429-13.9-c1-0-3
Degree $2$
Conductor $429$
Sign $-0.605 - 0.795i$
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  + (−0.333 + 0.578i)2-s + (−0.5 + 0.866i)3-s + (0.777 + 1.34i)4-s + 0.849·5-s + (−0.333 − 0.578i)6-s + (1.61 + 2.79i)7-s − 2.37·8-s + (−0.499 − 0.866i)9-s + (−0.283 + 0.490i)10-s + (0.5 − 0.866i)11-s − 1.55·12-s + (2.86 + 2.18i)13-s − 2.15·14-s + (−0.424 + 0.735i)15-s + (−0.762 + 1.32i)16-s + (0.394 + 0.683i)17-s + ⋯
L(s)  = 1  + (−0.236 + 0.408i)2-s + (−0.288 + 0.499i)3-s + (0.388 + 0.673i)4-s + 0.379·5-s + (−0.136 − 0.236i)6-s + (0.610 + 1.05i)7-s − 0.838·8-s + (−0.166 − 0.288i)9-s + (−0.0896 + 0.155i)10-s + (0.150 − 0.261i)11-s − 0.448·12-s + (0.795 + 0.606i)13-s − 0.576·14-s + (−0.109 + 0.189i)15-s + (−0.190 + 0.330i)16-s + (0.0956 + 0.165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581867 + 1.17475i\)
\(L(\frac12)\) \(\approx\) \(0.581867 + 1.17475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.86 - 2.18i)T \)
good2 \( 1 + (0.333 - 0.578i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.849T + 5T^{2} \)
7 \( 1 + (-1.61 - 2.79i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.394 - 0.683i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.49 + 2.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.75 + 3.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.576 - 0.998i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.0322T + 31T^{2} \)
37 \( 1 + (2.53 - 4.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 3.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.10 - 1.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.56T + 47T^{2} \)
53 \( 1 - 0.836T + 53T^{2} \)
59 \( 1 + (4.95 + 8.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.25 - 5.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.97 + 8.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.15 + 3.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.19T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (-4.75 + 8.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.84 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.49829363478726309086662593340, −10.74979165104961578046266872621, −9.380709855011164389711919105600, −8.751474534902907363270586520075, −8.020091625956454254578532449506, −6.62961578152440419263512139428, −5.98866647145829819409929284765, −4.83662692503042409393283588191, −3.48523964009994804775956055697, −2.14038996312803542883191918200, 0.986903610854775665953142493778, 2.04546535272199327544069989774, 3.74816904162277464183065581411, 5.26586307239486690266996256371, 6.09260860545516295761545757440, 7.10375777084580595468734885463, 7.991265617932282408215125981505, 9.223592346428917925626259223051, 10.25353599950128583462880996499, 10.77588748700267947709951430275

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