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Properties

Label 2-429-13.9-c1-0-5
Degree $2$
Conductor $429$
Sign $0.997 + 0.0713i$
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  + (0.423 − 0.734i)2-s + (0.5 − 0.866i)3-s + (0.640 + 1.10i)4-s − 2.90·5-s + (−0.423 − 0.734i)6-s + (2.22 + 3.84i)7-s + 2.78·8-s + (−0.499 − 0.866i)9-s + (−1.23 + 2.13i)10-s + (0.5 − 0.866i)11-s + 1.28·12-s + (3.53 − 0.704i)13-s + 3.76·14-s + (−1.45 + 2.51i)15-s + (−0.101 + 0.176i)16-s + (1.01 + 1.76i)17-s + ⋯
L(s)  = 1  + (0.299 − 0.519i)2-s + (0.288 − 0.499i)3-s + (0.320 + 0.554i)4-s − 1.30·5-s + (−0.173 − 0.299i)6-s + (0.839 + 1.45i)7-s + 0.983·8-s + (−0.166 − 0.288i)9-s + (−0.390 + 0.675i)10-s + (0.150 − 0.261i)11-s + 0.369·12-s + (0.980 − 0.195i)13-s + 1.00·14-s + (−0.375 + 0.650i)15-s + (−0.0254 + 0.0441i)16-s + (0.246 + 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.997 + 0.0713i$
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: \((2,\ 429,\ (\ :1/2),\ 0.997 + 0.0713i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81440 - 0.0648441i\)
\(L(\frac12)\) \(\approx\) \(1.81440 - 0.0648441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.53 + 0.704i)T \)
good2 \( 1 + (-0.423 + 0.734i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 + (-2.22 - 3.84i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-1.01 - 1.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.49 - 2.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.31 + 5.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.25 - 3.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 + (-3.33 + 5.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.03 - 5.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.57 + 9.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 + 1.52T + 53T^{2} \)
59 \( 1 + (-1.22 - 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.64 - 2.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.81 + 8.33i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 - 9.34T + 79T^{2} \)
83 \( 1 - 8.98T + 83T^{2} \)
89 \( 1 + (3.77 - 6.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.512 - 0.887i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.30288026941556342809457547516, −10.84147572533614112383724086332, −8.956042692001875095712589700231, −8.249454284945246535460090961207, −7.80617467593525867564441080441, −6.57606506421772721799989424243, −5.24935002824970654435268707305, −3.91242977133123305272792746084, −3.06214345676413057812898557840, −1.73738014302822584571026151566, 1.24929788799935002261252554063, 3.53875950965973326743581053137, 4.36412299651948363080453650631, 5.15557744219333325523995478445, 6.69806432832186751079738188686, 7.54458296320787657428163834838, 8.024309140731433092438415754752, 9.419957999045586666715751752250, 10.44630084256394747396560530548, 11.37377271610452689935896444989

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