Properties

Label 2-429-13.10-c1-0-19
Degree $2$
Conductor $429$
Sign $0.830 + 0.557i$
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  + (1.30 − 0.751i)2-s + (0.5 + 0.866i)3-s + (0.129 − 0.224i)4-s − 4.13i·5-s + (1.30 + 0.751i)6-s + (3.62 + 2.09i)7-s + 2.61i·8-s + (−0.499 + 0.866i)9-s + (−3.10 − 5.38i)10-s + (0.866 − 0.5i)11-s + 0.259·12-s + (−3.01 − 1.97i)13-s + 6.28·14-s + (3.58 − 2.06i)15-s + (2.22 + 3.85i)16-s + (2.36 − 4.10i)17-s + ⋯
L(s)  = 1  + (0.920 − 0.531i)2-s + (0.288 + 0.499i)3-s + (0.0648 − 0.112i)4-s − 1.84i·5-s + (0.531 + 0.306i)6-s + (1.36 + 0.790i)7-s + 0.925i·8-s + (−0.166 + 0.288i)9-s + (−0.982 − 1.70i)10-s + (0.261 − 0.150i)11-s + 0.0748·12-s + (−0.837 − 0.546i)13-s + 1.67·14-s + (0.924 − 0.533i)15-s + (0.556 + 0.963i)16-s + (0.574 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.830 + 0.557i$
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: \((2,\ 429,\ (\ :1/2),\ 0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42860 - 0.739487i\)
\(L(\frac12)\) \(\approx\) \(2.42860 - 0.739487i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.01 + 1.97i)T \)
good2 \( 1 + (-1.30 + 0.751i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 4.13iT - 5T^{2} \)
7 \( 1 + (-3.62 - 2.09i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-2.36 + 4.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 - 1.65i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.40 + 5.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.07 - 7.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.72iT - 31T^{2} \)
37 \( 1 + (2.97 - 1.71i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.94 - 4.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.783 - 1.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.637iT - 47T^{2} \)
53 \( 1 + 8.14T + 53T^{2} \)
59 \( 1 + (-2.80 - 1.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.37 - 4.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.59 - 2.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.20 + 5.31i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.37iT - 73T^{2} \)
79 \( 1 + 0.446T + 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + (-8.71 + 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.92 + 4.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.55599823493875638749488743278, −10.23618938766071153449933438691, −9.069209370157185672616671344556, −8.448283339552592226868626579964, −7.86116241849622436435876005602, −5.53299547528472785348514078751, −4.93777052994460096078863135884, −4.57508179511391001289702110050, −3.05084773698789196282753862850, −1.63077367873572093121865304777, 1.87673105718247249079128935782, 3.42647384501080873815516517233, 4.33696544761703823871753004165, 5.66410324766545825730028480302, 6.64116191380670640000622056229, 7.37142618237860887482032388733, 7.891274448317489315027823659919, 9.720300392044126464873897706342, 10.36727356834048174326547234804, 11.43436551911736520000531384105

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