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Properties

Label 2-429-13.12-c1-0-7
Degree $2$
Conductor $429$
Sign $0.185 - 0.982i$
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  + 2.25i·2-s − 3-s − 3.10·4-s − 3.76i·5-s − 2.25i·6-s + 0.701i·7-s − 2.48i·8-s + 9-s + 8.51·10-s + i·11-s + 3.10·12-s + (3.54 + 0.668i)13-s − 1.58·14-s + 3.76i·15-s − 0.589·16-s + 7.30·17-s + ⋯
L(s)  = 1  + 1.59i·2-s − 0.577·3-s − 1.55·4-s − 1.68i·5-s − 0.921i·6-s + 0.265i·7-s − 0.878i·8-s + 0.333·9-s + 2.69·10-s + 0.301i·11-s + 0.894·12-s + (0.982 + 0.185i)13-s − 0.423·14-s + 0.973i·15-s − 0.147·16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.185 - 0.982i$
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: \((2,\ 429,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891852 + 0.739214i\)
\(L(\frac12)\) \(\approx\) \(0.891852 + 0.739214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.54 - 0.668i)T \)
good2 \( 1 - 2.25iT - 2T^{2} \)
5 \( 1 + 3.76iT - 5T^{2} \)
7 \( 1 - 0.701iT - 7T^{2} \)
17 \( 1 - 7.30T + 17T^{2} \)
19 \( 1 - 3.97iT - 19T^{2} \)
23 \( 1 - 5.91T + 23T^{2} \)
29 \( 1 - 9.62T + 29T^{2} \)
31 \( 1 + 3.40iT - 31T^{2} \)
37 \( 1 + 9.91iT - 37T^{2} \)
41 \( 1 - 1.85iT - 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 3.39iT - 47T^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 + 1.08iT - 59T^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 - 4.26iT - 73T^{2} \)
79 \( 1 + 1.64T + 79T^{2} \)
83 \( 1 - 3.23iT - 83T^{2} \)
89 \( 1 - 3.20iT - 89T^{2} \)
97 \( 1 - 13.0iT - 97T^{2} \)
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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

−11.63617451513531670462715649639, −10.13963989962004607463281788161, −9.164113078903953843330895025060, −8.418167117509449952781346860926, −7.75640259791552860056890526939, −6.53676761692636907022587661106, −5.55684759797922647702866483750, −5.14129638720158680993493970243, −4.02619663983817232262918021724, −1.12536854442709461237003449264, 1.16969636050700195452304447552, 2.99208566597663772951707134475, 3.36360959962680807260636521118, 4.87016265959201913239328814025, 6.30051393173092880871738527291, 7.08171389696417439983008630355, 8.441220632184230597174931839527, 9.825729755853848873485066678258, 10.38786170009849754780877744755, 10.96005855906241940059317179084

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