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Properties

Label 2-429-11.4-c1-0-7
Degree $2$
Conductor $429$
Sign $-0.279 - 0.960i$
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  + (1.18 + 0.857i)2-s + (−0.309 + 0.951i)3-s + (0.0397 + 0.122i)4-s + (−0.326 + 0.237i)5-s + (−1.18 + 0.857i)6-s + (1.48 + 4.57i)7-s + (0.843 − 2.59i)8-s + (−0.809 − 0.587i)9-s − 0.589·10-s + (0.851 + 3.20i)11-s − 0.128·12-s + (−0.809 − 0.587i)13-s + (−2.16 + 6.67i)14-s + (−0.124 − 0.384i)15-s + (3.43 − 2.49i)16-s + (−2.90 + 2.11i)17-s + ⋯
L(s)  = 1  + (0.834 + 0.606i)2-s + (−0.178 + 0.549i)3-s + (0.0198 + 0.0612i)4-s + (−0.146 + 0.106i)5-s + (−0.481 + 0.350i)6-s + (0.561 + 1.72i)7-s + (0.298 − 0.918i)8-s + (−0.269 − 0.195i)9-s − 0.186·10-s + (0.256 + 0.966i)11-s − 0.0371·12-s + (−0.224 − 0.163i)13-s + (−0.579 + 1.78i)14-s + (−0.0322 − 0.0991i)15-s + (0.857 − 0.623i)16-s + (−0.705 + 0.512i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.279 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16689 + 1.55535i\)
\(L(\frac12)\) \(\approx\) \(1.16689 + 1.55535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.851 - 3.20i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.18 - 0.857i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.326 - 0.237i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.48 - 4.57i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (2.90 - 2.11i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.212 + 0.653i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + (0.367 + 1.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.30 + 0.946i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.26 - 3.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.78 + 11.6i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (-4.06 + 12.5i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.90 + 1.38i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.83 + 8.73i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.32 + 6.77i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + (3.86 - 2.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.21 + 6.81i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.00 - 3.63i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.44 - 1.77i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.70T + 89T^{2} \)
97 \( 1 + (-14.5 - 10.5i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.58183575939397375087642178554, −10.60020926147199653765662845439, −9.455481508215387837163878726686, −8.881015418746889454404145656358, −7.54069856990955050650541459380, −6.45666943870738369037804038954, −5.47820987717068113023412481634, −4.94061007913765744289470023046, −3.83366996854770768019010079335, −2.19836191766473819186680022767, 1.08290685167132651228799196843, 2.77124594621882059974852813729, 4.06069618044626361055474788400, 4.68222363529433487189858602772, 6.04739560503266874693146615135, 7.29157885698376817122485865637, 7.905816549740501278213045200941, 9.007299252752851958316165667805, 10.53709943792625562315799153549, 11.13911467364713504615142397392

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