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Properties

Label 2-429-33.32-c1-0-30
Degree $2$
Conductor $429$
Sign $-0.921 + 0.389i$
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  − 2.70·2-s + (−1.68 + 0.380i)3-s + 5.32·4-s − 2.40i·5-s + (4.57 − 1.02i)6-s − 4.64i·7-s − 8.98·8-s + (2.71 − 1.28i)9-s + 6.49i·10-s + (−0.588 − 3.26i)11-s + (−8.98 + 2.02i)12-s − i·13-s + 12.5i·14-s + (0.913 + 4.05i)15-s + 13.6·16-s + 2.96·17-s + ⋯
L(s)  = 1  − 1.91·2-s + (−0.975 + 0.219i)3-s + 2.66·4-s − 1.07i·5-s + (1.86 − 0.420i)6-s − 1.75i·7-s − 3.17·8-s + (0.903 − 0.428i)9-s + 2.05i·10-s + (−0.177 − 0.984i)11-s + (−2.59 + 0.584i)12-s − 0.277i·13-s + 3.36i·14-s + (0.235 + 1.04i)15-s + 3.41·16-s + 0.720·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.921 + 0.389i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.921 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0667513 - 0.329307i\)
\(L(\frac12)\) \(\approx\) \(0.0667513 - 0.329307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.380i)T \)
11 \( 1 + (0.588 + 3.26i)T \)
13 \( 1 + iT \)
good2 \( 1 + 2.70T + 2T^{2} \)
5 \( 1 + 2.40iT - 5T^{2} \)
7 \( 1 + 4.64iT - 7T^{2} \)
17 \( 1 - 2.96T + 17T^{2} \)
19 \( 1 - 0.907iT - 19T^{2} \)
23 \( 1 + 0.745iT - 23T^{2} \)
29 \( 1 - 4.15T + 29T^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 + 6.71iT - 43T^{2} \)
47 \( 1 - 6.35iT - 47T^{2} \)
53 \( 1 + 6.81iT - 53T^{2} \)
59 \( 1 + 5.61iT - 59T^{2} \)
61 \( 1 - 5.76iT - 61T^{2} \)
67 \( 1 - 8.08T + 67T^{2} \)
71 \( 1 - 0.0134iT - 71T^{2} \)
73 \( 1 - 4.80iT - 73T^{2} \)
79 \( 1 + 8.17iT - 79T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 - 6.88iT - 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−10.51201284218259694126734450778, −10.06161712132043683708232460980, −9.023496602951390063238189245670, −8.098524368281948837314912104950, −7.30857116375535515408828819174, −6.39172394418179647228676308081, −5.21343186102819086752544835391, −3.65135609648423251447147542317, −1.22963361552281069764041908356, −0.51020457912995216038073863695, 1.80884225979324305991625925376, 2.78925846469621921937569798248, 5.37677627013305385104660164620, 6.35492575907896280058103958609, 7.01737864711306954176244415680, 7.88676543243407505814022027377, 8.990048618668363482652280307161, 9.821774325119245685756063572747, 10.50905470799007820703852429738, 11.32732269800649216926113849788

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