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Properties

Label 2-429-13.9-c1-0-10
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: $\chi_{429} (100, \cdot )$
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

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

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