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Properties

Label 2-429-13.12-c1-0-22
Degree $2$
Conductor $429$
Sign $-0.489 - 0.872i$
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  − 2.35i·2-s − 3-s − 3.56·4-s − 1.80i·5-s + 2.35i·6-s − 4.78i·7-s + 3.70i·8-s + 9-s − 4.25·10-s + i·11-s + 3.56·12-s + (3.14 − 1.76i)13-s − 11.2·14-s + 1.80i·15-s + 1.59·16-s − 2.63·17-s + ⋯
L(s)  = 1  − 1.66i·2-s − 0.577·3-s − 1.78·4-s − 0.806i·5-s + 0.963i·6-s − 1.80i·7-s + 1.30i·8-s + 0.333·9-s − 1.34·10-s + 0.301i·11-s + 1.03·12-s + (0.872 − 0.489i)13-s − 3.01·14-s + 0.465i·15-s + 0.398·16-s − 0.639·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.489 - 0.872i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440202 + 0.751781i\)
\(L(\frac12)\) \(\approx\) \(0.440202 + 0.751781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 - iT \)
13 \( 1 + (-3.14 + 1.76i)T \)
good2 \( 1 + 2.35iT - 2T^{2} \)
5 \( 1 + 1.80iT - 5T^{2} \)
7 \( 1 + 4.78iT - 7T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 6.47iT - 19T^{2} \)
23 \( 1 + 2.40T + 23T^{2} \)
29 \( 1 + 1.40T + 29T^{2} \)
31 \( 1 - 6.06iT - 31T^{2} \)
37 \( 1 + 6.63iT - 37T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 - 7.35T + 43T^{2} \)
47 \( 1 + 9.35iT - 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 + 0.288iT - 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 - 8.88iT - 71T^{2} \)
73 \( 1 + 2.61iT - 73T^{2} \)
79 \( 1 - 3.32T + 79T^{2} \)
83 \( 1 - 9.78iT - 83T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
97 \( 1 - 4.40iT - 97T^{2} \)
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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

−10.56709410242191124899677591048, −10.20526227338470398005299865574, −9.105366607639628553000968606065, −8.035164422363954730593260009594, −6.82128589558156811184110346820, −5.32754231761749783515340370634, −4.17158962565685458513104288677, −3.69794256449030610226289720639, −1.65488555039638786718257643354, −0.63057211307555144323019821253, 2.63037277937520789332548578160, 4.47237090202087171193286166930, 5.52435059930843055449935507077, 6.27685045014212678567082073179, 6.74066478991890273316352866420, 8.004801378768121834435626254407, 8.865368719000561931618974002703, 9.496896608544201712396178114113, 11.11919336728219130004787294490, 11.54345673721374454051992866915

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