Properties

Label 2-429-13.12-c1-0-19
Degree $2$
Conductor $429$
Sign $-0.920 + 0.390i$
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  − 2.15i·2-s + 3-s − 2.65·4-s + 0.710i·5-s − 2.15i·6-s − 2.30i·7-s + 1.41i·8-s + 9-s + 1.53·10-s i·11-s − 2.65·12-s + (−1.40 − 3.31i)13-s − 4.98·14-s + 0.710i·15-s − 2.26·16-s + 6.68·17-s + ⋯
L(s)  = 1  − 1.52i·2-s + 0.577·3-s − 1.32·4-s + 0.317i·5-s − 0.880i·6-s − 0.872i·7-s + 0.499i·8-s + 0.333·9-s + 0.484·10-s − 0.301i·11-s − 0.766·12-s + (−0.390 − 0.920i)13-s − 1.33·14-s + 0.183i·15-s − 0.565·16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.920 + 0.390i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.920 + 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306795 - 1.50832i\)
\(L(\frac12)\) \(\approx\) \(0.306795 - 1.50832i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (1.40 + 3.31i)T \)
good2 \( 1 + 2.15iT - 2T^{2} \)
5 \( 1 - 0.710iT - 5T^{2} \)
7 \( 1 + 2.30iT - 7T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
19 \( 1 + 0.242iT - 19T^{2} \)
23 \( 1 + 9.53T + 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 + 4.02iT - 31T^{2} \)
37 \( 1 + 3.42iT - 37T^{2} \)
41 \( 1 - 9.46iT - 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 6.78T + 53T^{2} \)
59 \( 1 + 7.81iT - 59T^{2} \)
61 \( 1 - 0.910T + 61T^{2} \)
67 \( 1 - 9.32iT - 67T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + 8.51iT - 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 - 4.64iT - 83T^{2} \)
89 \( 1 + 14.7iT - 89T^{2} \)
97 \( 1 - 1.86iT - 97T^{2} \)
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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

−10.63819200076029224176108870419, −10.09416721917286360207037299548, −9.437010423716836493374209821300, −8.078046065507233250251146280468, −7.39605571768039315499682580542, −5.84956397017149358300851327369, −4.30077907495101493020147361071, −3.45970106193752902788527104181, −2.53552748306959786824467233407, −0.986421031472242657310553305087, 2.18894990468798976168821848605, 3.94565013479149826872982978235, 5.16461496068806521156812932626, 5.91241915442037344969878480136, 7.04817298566896246742329963810, 7.80674350154646464541328252285, 8.671690787033310816037542488489, 9.283348073045706863180492904258, 10.27691972112797558355156582694, 11.96355402433043998717128109827

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