Properties

Label 2-429-13.3-c1-0-5
Degree $2$
Conductor $429$
Sign $-0.547 - 0.836i$
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  + (0.893 + 1.54i)2-s + (−0.5 − 0.866i)3-s + (−0.596 + 1.03i)4-s − 0.535·5-s + (0.893 − 1.54i)6-s + (−2.30 + 4.00i)7-s + 1.44·8-s + (−0.499 + 0.866i)9-s + (−0.478 − 0.828i)10-s + (0.5 + 0.866i)11-s + 1.19·12-s + (1.10 + 3.43i)13-s − 8.25·14-s + (0.267 + 0.463i)15-s + (2.48 + 4.29i)16-s + (−2.01 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.631 + 1.09i)2-s + (−0.288 − 0.499i)3-s + (−0.298 + 0.516i)4-s − 0.239·5-s + (0.364 − 0.631i)6-s + (−0.873 + 1.51i)7-s + 0.510·8-s + (−0.166 + 0.288i)9-s + (−0.151 − 0.261i)10-s + (0.150 + 0.261i)11-s + 0.344·12-s + (0.306 + 0.951i)13-s − 2.20·14-s + (0.0691 + 0.119i)15-s + (0.620 + 1.07i)16-s + (−0.487 + 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.547 - 0.836i$
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: \((2,\ 429,\ (\ :1/2),\ -0.547 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732960 + 1.35584i\)
\(L(\frac12)\) \(\approx\) \(0.732960 + 1.35584i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 - 3.43i)T \)
good2 \( 1 + (-0.893 - 1.54i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.535T + 5T^{2} \)
7 \( 1 + (2.30 - 4.00i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.01 - 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.75 + 6.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 4.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.41 - 4.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (1.99 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.70 + 8.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.90 + 6.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.31T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 + (2.07 - 3.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.87 + 8.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.91 + 6.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.18 - 8.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.96T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 7.36T + 83T^{2} \)
89 \( 1 + (1.68 + 2.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.95 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.75606603129439762074779438814, −10.75543897736799951446535046877, −9.267135539858140620471250767003, −8.717860452729443731183991775848, −7.29803117578175410413597845955, −6.76542887564344017107013601900, −5.82164400169444378165228400232, −5.18263584690916710978702469556, −3.75247959900953949793907612644, −2.09352299535168126887637281681, 0.870127760368795733359959870770, 3.00815892927743576940307678187, 3.75217313228270146566772201052, 4.54664674471321447733554884090, 5.86429573105328217546581628879, 7.11888564277233345256275437494, 8.029779708074334360761521035798, 9.639368800653042881821067443321, 10.24061106879311022428638020516, 10.84153052521485128308859830182

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