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Properties

Label 2-429-13.3-c1-0-17
Degree $2$
Conductor $429$
Sign $0.964 + 0.263i$
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  + (0.362 + 0.627i)2-s + (−0.5 − 0.866i)3-s + (0.737 − 1.27i)4-s + 4.26·5-s + (0.362 − 0.627i)6-s + (−0.335 + 0.580i)7-s + 2.51·8-s + (−0.499 + 0.866i)9-s + (1.54 + 2.67i)10-s + (0.5 + 0.866i)11-s − 1.47·12-s + (−3.10 − 1.83i)13-s − 0.486·14-s + (−2.13 − 3.69i)15-s + (−0.561 − 0.972i)16-s + (−2.17 + 3.76i)17-s + ⋯
L(s)  = 1  + (0.256 + 0.443i)2-s + (−0.288 − 0.499i)3-s + (0.368 − 0.638i)4-s + 1.90·5-s + (0.147 − 0.256i)6-s + (−0.126 + 0.219i)7-s + 0.890·8-s + (−0.166 + 0.288i)9-s + (0.489 + 0.847i)10-s + (0.150 + 0.261i)11-s − 0.425·12-s + (−0.860 − 0.509i)13-s − 0.129·14-s + (−0.551 − 0.954i)15-s + (−0.140 − 0.243i)16-s + (−0.526 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.964 + 0.263i$
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: \((2,\ 429,\ (\ :1/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04861 - 0.275046i\)
\(L(\frac12)\) \(\approx\) \(2.04861 - 0.275046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.10 + 1.83i)T \)
good2 \( 1 + (-0.362 - 0.627i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 4.26T + 5T^{2} \)
7 \( 1 + (0.335 - 0.580i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.17 - 3.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.19 - 2.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.47 + 6.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.22 - 5.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.82T + 31T^{2} \)
37 \( 1 + (-2.73 - 4.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.32 + 10.9i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.08T + 47T^{2} \)
53 \( 1 + 5.80T + 53T^{2} \)
59 \( 1 + (-4.64 + 8.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.685 + 1.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.32 + 7.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.77 - 8.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.15T + 73T^{2} \)
79 \( 1 - 3.19T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + (-3.17 - 5.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.76 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76649564911596359462958134349, −10.31374578580600610458486614517, −9.525573921372353329741083932798, −8.346340133402506114719629800936, −6.91329151989659011925950360462, −6.34803324913879083317893753339, −5.63089436787654253665209732889, −4.81653287953960511469480327446, −2.44398802009803772231881901564, −1.61528224974801848834892973885, 1.92027152544176054347169860564, 2.88570223564764595638526421841, 4.36090142881107509828885703356, 5.37095332444539286905520889853, 6.42604837551732532028071757374, 7.27094049490661837933720474471, 8.801407578690528071289447866839, 9.683144985138433450678793003765, 10.20060284238094384664510420085, 11.26374288732956082297699707814

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