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Properties

Label 2-429-11.5-c1-0-11
Degree $2$
Conductor $429$
Sign $0.745 - 0.666i$
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  + (0.0974 + 0.299i)2-s + (0.809 + 0.587i)3-s + (1.53 − 1.11i)4-s + (−0.824 + 2.53i)5-s + (−0.0974 + 0.299i)6-s + (1.23 − 0.894i)7-s + (0.995 + 0.722i)8-s + (0.309 + 0.951i)9-s − 0.841·10-s + (1.53 − 2.94i)11-s + 1.90·12-s + (0.309 + 0.951i)13-s + (0.388 + 0.281i)14-s + (−2.15 + 1.56i)15-s + (1.05 − 3.24i)16-s + (−1.57 + 4.84i)17-s + ⋯
L(s)  = 1  + (0.0689 + 0.212i)2-s + (0.467 + 0.339i)3-s + (0.768 − 0.558i)4-s + (−0.368 + 1.13i)5-s + (−0.0397 + 0.122i)6-s + (0.465 − 0.337i)7-s + (0.351 + 0.255i)8-s + (0.103 + 0.317i)9-s − 0.265·10-s + (0.462 − 0.886i)11-s + 0.548·12-s + (0.0857 + 0.263i)13-s + (0.103 + 0.0753i)14-s + (−0.557 + 0.404i)15-s + (0.263 − 0.811i)16-s + (−0.381 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85149 + 0.707316i\)
\(L(\frac12)\) \(\approx\) \(1.85149 + 0.707316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.53 + 2.94i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.0974 - 0.299i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.824 - 2.53i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.23 + 0.894i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (1.57 - 4.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.41 - 1.75i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 + (-0.565 + 0.411i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.54 + 4.74i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.16 + 5.20i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.53 - 5.47i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + (8.78 + 6.38i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.163 - 0.504i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.75 + 7.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.462 + 1.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (-1.34 + 4.12i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.98 - 4.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.68 + 5.17i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.51 + 7.73i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.51T + 89T^{2} \)
97 \( 1 + (0.538 + 1.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.24427640155959162819242235821, −10.46674511757798988912797121026, −9.702553679643271311350555938942, −8.274875572085523690052741553732, −7.60851141852911478190256773128, −6.53242280512971271201660611779, −5.84067069084662894961268837531, −4.22434263599438043944671689017, −3.21548656460422701091806740133, −1.84820614265868228094203688224, 1.47244438948870785775653498605, 2.71004791365280269434158268502, 4.10526304948691798658852462948, 5.07197371262150487031805282780, 6.59053554325608717635641945426, 7.52913190573431477893479140767, 8.244744652753759933134244513018, 9.081152583976852440770734373200, 10.07871762315648695798344082217, 11.59603896508628840755628260556

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