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Properties

Label 2-429-33.32-c1-0-44
Degree $2$
Conductor $429$
Sign $-0.658 + 0.752i$
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  + 1.67·2-s + (−1.56 − 0.747i)3-s + 0.796·4-s − 2.44i·5-s + (−2.61 − 1.25i)6-s − 0.956i·7-s − 2.01·8-s + (1.88 + 2.33i)9-s − 4.08i·10-s + (−3.19 − 0.890i)11-s + (−1.24 − 0.595i)12-s − i·13-s − 1.60i·14-s + (−1.82 + 3.81i)15-s − 4.95·16-s − 2.79·17-s + ⋯
L(s)  = 1  + 1.18·2-s + (−0.902 − 0.431i)3-s + 0.398·4-s − 1.09i·5-s + (−1.06 − 0.510i)6-s − 0.361i·7-s − 0.711·8-s + (0.627 + 0.778i)9-s − 1.29i·10-s + (−0.963 − 0.268i)11-s + (−0.359 − 0.171i)12-s − 0.277i·13-s − 0.427i·14-s + (−0.471 + 0.985i)15-s − 1.23·16-s − 0.677·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523334 - 1.15243i\)
\(L(\frac12)\) \(\approx\) \(0.523334 - 1.15243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.747i)T \)
11 \( 1 + (3.19 + 0.890i)T \)
13 \( 1 + iT \)
good2 \( 1 - 1.67T + 2T^{2} \)
5 \( 1 + 2.44iT - 5T^{2} \)
7 \( 1 + 0.956iT - 7T^{2} \)
17 \( 1 + 2.79T + 17T^{2} \)
19 \( 1 + 1.65iT - 19T^{2} \)
23 \( 1 + 7.92iT - 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 - 6.88T + 31T^{2} \)
37 \( 1 + 2.79T + 37T^{2} \)
41 \( 1 - 5.64T + 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + 6.25iT - 47T^{2} \)
53 \( 1 - 7.02iT - 53T^{2} \)
59 \( 1 + 6.12iT - 59T^{2} \)
61 \( 1 + 1.79iT - 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 8.11iT - 71T^{2} \)
73 \( 1 - 8.25iT - 73T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 5.15iT - 89T^{2} \)
97 \( 1 - 7.74T + 97T^{2} \)
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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

−11.07409740708229473208158848704, −10.26175045898607962660917762186, −8.883547533048429013040164655492, −8.005711934451720148213588732794, −6.66687938492804887535793081901, −5.86774973435490894835418018385, −4.74255885619686720190030435270, −4.56621784590226074844068560050, −2.67275172562617123583162145695, −0.60463317423160887890613390225, 2.61409705000048734825368326242, 3.73537061977987653226773922524, 4.79783906248949341969496247551, 5.65748843999255432601484428202, 6.44751335420717934128027485217, 7.36730007275834167639115204055, 8.943535752841651216179591389148, 10.04201239863049509069057120636, 10.76322720412790011469347858584, 11.67541025236259745970304842872

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