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Properties

Label 2-429-429.428-c1-0-33
Degree $2$
Conductor $429$
Sign $-0.438 + 0.898i$
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  − 2.63i·2-s + (1.55 + 0.758i)3-s − 4.93·4-s + (1.99 − 4.10i)6-s + 4.70·7-s + 7.72i·8-s + (1.84 + 2.36i)9-s − 3.31i·11-s + (−7.68 − 3.74i)12-s + 3.60·13-s − 12.3i·14-s + 10.4·16-s + (6.22 − 4.86i)18-s − 7.09·19-s + (7.32 + 3.56i)21-s − 8.73·22-s + ⋯
L(s)  = 1  − 1.86i·2-s + (0.898 + 0.438i)3-s − 2.46·4-s + (0.815 − 1.67i)6-s + 1.77·7-s + 2.73i·8-s + (0.616 + 0.787i)9-s − 1.00i·11-s + (−2.21 − 1.08i)12-s + 1.00·13-s − 3.31i·14-s + 2.61·16-s + (1.46 − 1.14i)18-s − 1.62·19-s + (1.59 + 0.778i)21-s − 1.86·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.438 + 0.898i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995798 - 1.59301i\)
\(L(\frac12)\) \(\approx\) \(0.995798 - 1.59301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.55 - 0.758i)T \)
11 \( 1 + 3.31iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 2.63iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 4.70T + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
23 \( 1 - 0.711iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12.1iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 17.1iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.85798097531876468010272527154, −10.39033914486739507475384501003, −8.998865330838072603828062747334, −8.602347600806744337524626196963, −7.87551963073472661192116839299, −5.55779584106842475777690892599, −4.36790873429826568790662462098, −3.79290745629044587948433860750, −2.42608125043533494531090970451, −1.47144112014320987995452947594, 1.75675770060327534331312720937, 4.09077564662709769520143162064, 4.74287917022030438899352947807, 6.06037382152240171722848163700, 6.96493375383921222033598772519, 7.927068599795247891349769346601, 8.269775668297176098546385062066, 9.032585814254404030299023358443, 10.16700982088128763655503231869, 11.55911003590685450539023101248

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