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Properties

Label 2-429-13.10-c1-0-9
Degree $2$
Conductor $429$
Sign $0.165 - 0.986i$
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  + (−1.91 + 1.10i)2-s + (0.5 + 0.866i)3-s + (1.45 − 2.52i)4-s − 0.692i·5-s + (−1.91 − 1.10i)6-s + (2.81 + 1.62i)7-s + 2.02i·8-s + (−0.499 + 0.866i)9-s + (0.767 + 1.32i)10-s + (−0.866 + 0.5i)11-s + 2.91·12-s + (0.552 − 3.56i)13-s − 7.19·14-s + (0.599 − 0.346i)15-s + (0.672 + 1.16i)16-s + (2.74 − 4.74i)17-s + ⋯
L(s)  = 1  + (−1.35 + 0.783i)2-s + (0.288 + 0.499i)3-s + (0.727 − 1.26i)4-s − 0.309i·5-s + (−0.783 − 0.452i)6-s + (1.06 + 0.613i)7-s + 0.714i·8-s + (−0.166 + 0.288i)9-s + (0.242 + 0.420i)10-s + (−0.261 + 0.150i)11-s + 0.840·12-s + (0.153 − 0.988i)13-s − 1.92·14-s + (0.154 − 0.0893i)15-s + (0.168 + 0.291i)16-s + (0.665 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.675648 + 0.571546i\)
\(L(\frac12)\) \(\approx\) \(0.675648 + 0.571546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.552 + 3.56i)T \)
good2 \( 1 + (1.91 - 1.10i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.692iT - 5T^{2} \)
7 \( 1 + (-2.81 - 1.62i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-2.74 + 4.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.07 - 5.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.313 + 0.542i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.11iT - 31T^{2} \)
37 \( 1 + (5.80 - 3.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.53 - 3.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.752 - 1.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 - 3.70T + 53T^{2} \)
59 \( 1 + (-3.35 - 1.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.33 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 6.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.44 - 2.56i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 + 9.59iT - 83T^{2} \)
89 \( 1 + (-11.9 + 6.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.75 + 1.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.07310493464993436444720785508, −10.06009297888234719765759301132, −9.438691382280836267467174519584, −8.533905520353884260431693166893, −7.938674237775070824558351898756, −7.18876892482293033276309260450, −5.59557053930392032998335507959, −5.02941431130482849302722087135, −3.13577507613480696733699547874, −1.28756581070607702107564336034, 1.09576367268707466364624132545, 2.16748244432590125570559692626, 3.52854275154957062215427349501, 5.10477420820963378822653988169, 6.82198452287683886733423482232, 7.52335073220775394299266460815, 8.483583460284053241254079544235, 8.959738359742784427105403390267, 10.29509569588210942136845205265, 10.73388916054334249978282607633

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