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Properties

Label 2-429-143.21-c1-0-1
Degree $2$
Conductor $429$
Sign $-0.996 + 0.0822i$
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  + (−1.92 + 1.92i)2-s − 3-s − 5.42i·4-s + (−1.59 − 1.59i)5-s + (1.92 − 1.92i)6-s + (−1.66 − 1.66i)7-s + (6.60 + 6.60i)8-s + 9-s + 6.16·10-s + (2.43 + 2.25i)11-s + 5.42i·12-s + (1.55 − 3.25i)13-s + 6.42·14-s + (1.59 + 1.59i)15-s − 14.6·16-s − 4.73·17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s − 0.577·3-s − 2.71i·4-s + (−0.714 − 0.714i)5-s + (0.786 − 0.786i)6-s + (−0.630 − 0.630i)7-s + (2.33 + 2.33i)8-s + 0.333·9-s + 1.94·10-s + (0.733 + 0.680i)11-s + 1.56i·12-s + (0.431 − 0.901i)13-s + 1.71·14-s + (0.412 + 0.412i)15-s − 3.65·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.996 + 0.0822i$
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: \((2,\ 429,\ (\ :1/2),\ -0.996 + 0.0822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00457382 - 0.111071i\)
\(L(\frac12)\) \(\approx\) \(0.00457382 - 0.111071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 - 2.25i)T \)
13 \( 1 + (-1.55 + 3.25i)T \)
good2 \( 1 + (1.92 - 1.92i)T - 2iT^{2} \)
5 \( 1 + (1.59 + 1.59i)T + 5iT^{2} \)
7 \( 1 + (1.66 + 1.66i)T + 7iT^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + (3.03 - 3.03i)T - 19iT^{2} \)
23 \( 1 + 1.62iT - 23T^{2} \)
29 \( 1 - 3.51iT - 29T^{2} \)
31 \( 1 + (-4.40 - 4.40i)T + 31iT^{2} \)
37 \( 1 + (3.16 - 3.16i)T - 37iT^{2} \)
41 \( 1 + (4.68 - 4.68i)T - 41iT^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 + (6.86 - 6.86i)T - 47iT^{2} \)
53 \( 1 + 0.475T + 53T^{2} \)
59 \( 1 + (-8.18 + 8.18i)T - 59iT^{2} \)
61 \( 1 - 9.65iT - 61T^{2} \)
67 \( 1 + (-5.61 - 5.61i)T + 67iT^{2} \)
71 \( 1 + (-5.96 - 5.96i)T + 71iT^{2} \)
73 \( 1 + (1.78 + 1.78i)T + 73iT^{2} \)
79 \( 1 + 1.07iT - 79T^{2} \)
83 \( 1 + (10.3 - 10.3i)T - 83iT^{2} \)
89 \( 1 + (13.0 - 13.0i)T - 89iT^{2} \)
97 \( 1 + (-8.39 - 8.39i)T + 97iT^{2} \)
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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

−11.25404743962142447997676809940, −10.34277497732713178069643208500, −9.752576426420972435287752507444, −8.571080042442405806963058260822, −8.145408582560558677037619663277, −6.78898167185258925082792999486, −6.58831684917931805832981719812, −5.20959387174715940453889491219, −4.22590243862982196836219169895, −1.23205317726776421776007713602, 0.13432560182281011238523785844, 2.05987407328431976015132522923, 3.33272058884851443195164568910, 4.21419616862869527540520938696, 6.43397949219690468305893153062, 7.06846304283847811517506151596, 8.417330967439417414674751811699, 9.019254280389355710686815098154, 9.885148193172902771465986400060, 10.92428480365270874897108246299

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