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Properties

Label 2-429-13.9-c1-0-12
Degree $2$
Conductor $429$
Sign $0.449 + 0.893i$
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  + (1.08 − 1.87i)2-s + (−0.5 + 0.866i)3-s + (−1.33 − 2.30i)4-s + 2.38·5-s + (1.08 + 1.87i)6-s + (1.17 + 2.04i)7-s − 1.43·8-s + (−0.499 − 0.866i)9-s + (2.57 − 4.46i)10-s + (−0.5 + 0.866i)11-s + 2.66·12-s + (2.41 + 2.67i)13-s + 5.08·14-s + (−1.19 + 2.06i)15-s + (1.11 − 1.92i)16-s + (−2.96 − 5.12i)17-s + ⋯
L(s)  = 1  + (0.763 − 1.32i)2-s + (−0.288 + 0.499i)3-s + (−0.666 − 1.15i)4-s + 1.06·5-s + (0.440 + 0.763i)6-s + (0.445 + 0.771i)7-s − 0.508·8-s + (−0.166 − 0.288i)9-s + (0.815 − 1.41i)10-s + (−0.150 + 0.261i)11-s + 0.769·12-s + (0.670 + 0.742i)13-s + 1.36·14-s + (−0.308 + 0.533i)15-s + (0.278 − 0.481i)16-s + (−0.717 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.449 + 0.893i$
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: \((2,\ 429,\ (\ :1/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93427 - 1.19196i\)
\(L(\frac12)\) \(\approx\) \(1.93427 - 1.19196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 2.67i)T \)
good2 \( 1 + (-1.08 + 1.87i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.38T + 5T^{2} \)
7 \( 1 + (-1.17 - 2.04i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (2.96 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 1.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.79 + 4.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.47 - 6.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 + (-0.535 + 0.926i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.75 + 8.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.28 - 2.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.85T + 47T^{2} \)
53 \( 1 - 1.27T + 53T^{2} \)
59 \( 1 + (-4.30 - 7.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.45 + 2.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.38 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.99 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 4.66T + 79T^{2} \)
83 \( 1 - 9.48T + 83T^{2} \)
89 \( 1 + (0.114 - 0.198i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.77 + 4.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.13593580808111963707494630151, −10.40114143303617269203633076881, −9.393184468867466954558858523377, −8.894635062388935333600895749603, −7.05081527892944578993366187636, −5.67901535808902183796673898880, −5.10658950690510745886556323548, −4.02797891498357656077510458566, −2.66275957613834706964150668583, −1.72114394163922397618604180997, 1.65282791833468586658194259345, 3.67933339413244206907840508528, 4.95401862165190614034681815125, 5.84703051239884319912308627175, 6.35782850396208822189587186767, 7.47739571542442937424939299462, 8.093477659375102300883826952558, 9.319153085519516466377801173707, 10.62195710723218480422032053230, 11.19949440416795647951413439667

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