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Properties

Label 2-429-13.10-c1-0-4
Degree $2$
Conductor $429$
Sign $-0.942 - 0.334i$
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  + (−2.12 + 1.22i)2-s + (−0.5 − 0.866i)3-s + (2.00 − 3.46i)4-s + 4.42i·5-s + (2.12 + 1.22i)6-s + (0.902 + 0.521i)7-s + 4.91i·8-s + (−0.499 + 0.866i)9-s + (−5.41 − 9.38i)10-s + (−0.866 + 0.5i)11-s − 4.00·12-s + (3.41 + 1.16i)13-s − 2.55·14-s + (3.82 − 2.21i)15-s + (−2.01 − 3.49i)16-s + (0.958 − 1.65i)17-s + ⋯
L(s)  = 1  + (−1.50 + 0.866i)2-s + (−0.288 − 0.499i)3-s + (1.00 − 1.73i)4-s + 1.97i·5-s + (0.866 + 0.500i)6-s + (0.341 + 0.196i)7-s + 1.73i·8-s + (−0.166 + 0.288i)9-s + (−1.71 − 2.96i)10-s + (−0.261 + 0.150i)11-s − 1.15·12-s + (0.946 + 0.322i)13-s − 0.682·14-s + (0.988 − 0.570i)15-s + (−0.504 − 0.873i)16-s + (0.232 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.942 - 0.334i$
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: \((2,\ 429,\ (\ :1/2),\ -0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0868418 + 0.504061i\)
\(L(\frac12)\) \(\approx\) \(0.0868418 + 0.504061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 - 1.16i)T \)
good2 \( 1 + (2.12 - 1.22i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 4.42iT - 5T^{2} \)
7 \( 1 + (-0.902 - 0.521i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.958 + 1.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.48 - 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 + 2.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 - 6.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.56iT - 31T^{2} \)
37 \( 1 + (7.19 - 4.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.57 - 0.911i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.06 - 5.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.92iT - 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + (-8.66 - 5.00i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.34 - 2.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.52 - 3.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.30 + 1.32i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.87iT - 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 5.54iT - 83T^{2} \)
89 \( 1 + (7.96 - 4.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.679 + 0.392i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.15265058730187743979975402323, −10.49639864319964767273290831454, −9.869625395556414759515961145607, −8.573231207360324745374651813426, −7.82100661473080863905481276810, −6.85215226106619198152114223702, −6.60219830415533834126872827076, −5.48783225251648551415845146870, −3.20132600843842582722331230068, −1.71657736689674117556904696065, 0.57813948539989326310944915580, 1.67664121701251321556672372399, 3.59819055718600131217982857450, 4.79437514472090411245435323143, 5.87869958780925686299005046654, 7.80077140498257815776589816402, 8.294784047467609095952554682141, 9.113559658191937645233769922080, 9.701539417439351010101353592189, 10.64404183442774122599124871231

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