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Properties

Label 2-429-143.21-c1-0-13
Degree $2$
Conductor $429$
Sign $0.724 - 0.689i$
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  + (0.290 − 0.290i)2-s + 3-s + 1.83i·4-s + (1.55 + 1.55i)5-s + (0.290 − 0.290i)6-s + (−0.786 − 0.786i)7-s + (1.11 + 1.11i)8-s + 9-s + 0.902·10-s + (−1.44 − 2.98i)11-s + 1.83i·12-s + (2.05 + 2.96i)13-s − 0.457·14-s + (1.55 + 1.55i)15-s − 3.01·16-s + 2.52·17-s + ⋯
L(s)  = 1  + (0.205 − 0.205i)2-s + 0.577·3-s + 0.915i·4-s + (0.694 + 0.694i)5-s + (0.118 − 0.118i)6-s + (−0.297 − 0.297i)7-s + (0.393 + 0.393i)8-s + 0.333·9-s + 0.285·10-s + (−0.434 − 0.900i)11-s + 0.528i·12-s + (0.568 + 0.822i)13-s − 0.122·14-s + (0.400 + 0.400i)15-s − 0.753·16-s + 0.613·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.724 - 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86715 + 0.746012i\)
\(L(\frac12)\) \(\approx\) \(1.86715 + 0.746012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 2.98i)T \)
13 \( 1 + (-2.05 - 2.96i)T \)
good2 \( 1 + (-0.290 + 0.290i)T - 2iT^{2} \)
5 \( 1 + (-1.55 - 1.55i)T + 5iT^{2} \)
7 \( 1 + (0.786 + 0.786i)T + 7iT^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 + (1.03 - 1.03i)T - 19iT^{2} \)
23 \( 1 - 5.04iT - 23T^{2} \)
29 \( 1 - 0.703iT - 29T^{2} \)
31 \( 1 + (2.73 + 2.73i)T + 31iT^{2} \)
37 \( 1 + (-1.75 + 1.75i)T - 37iT^{2} \)
41 \( 1 + (-7.42 + 7.42i)T - 41iT^{2} \)
43 \( 1 + 9.78T + 43T^{2} \)
47 \( 1 + (-2.29 + 2.29i)T - 47iT^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + (2.75 - 2.75i)T - 59iT^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + (9.79 + 9.79i)T + 67iT^{2} \)
71 \( 1 + (2.26 + 2.26i)T + 71iT^{2} \)
73 \( 1 + (0.116 + 0.116i)T + 73iT^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 + (-4.15 + 4.15i)T - 83iT^{2} \)
89 \( 1 + (2.00 - 2.00i)T - 89iT^{2} \)
97 \( 1 + (3.65 + 3.65i)T + 97iT^{2} \)
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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

−11.23207185764022684883927919153, −10.45164771886389865397794990613, −9.424483126830909463806714847752, −8.536823184587154630562585291819, −7.61649545275641782196214881522, −6.72161076700478302239131141832, −5.60698065611421299343687478124, −3.96318278281593630688827968819, −3.23778201728696265363471961702, −2.09951610268075847427435124701, 1.33186036236133193649226946182, 2.68437783236168623806324385809, 4.40238007425670355475312770716, 5.35118365789433148279942952075, 6.14005441790301618207284770224, 7.29592581667873661205667841276, 8.475574343272906506382289783553, 9.334283147542893447963409300914, 10.04570059263907868482439803564, 10.71453126916364194873086024381

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