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Properties

Label 2-429-13.10-c1-0-17
Degree $2$
Conductor $429$
Sign $0.291 + 0.956i$
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  + (−1.57 + 0.908i)2-s + (−0.5 − 0.866i)3-s + (0.649 − 1.12i)4-s − 0.705i·5-s + (1.57 + 0.908i)6-s + (1.35 + 0.784i)7-s − 1.27i·8-s + (−0.499 + 0.866i)9-s + (0.641 + 1.11i)10-s + (−0.866 + 0.5i)11-s − 1.29·12-s + (−1.09 − 3.43i)13-s − 2.84·14-s + (−0.611 + 0.352i)15-s + (2.45 + 4.25i)16-s + (−1.66 + 2.88i)17-s + ⋯
L(s)  = 1  + (−1.11 + 0.642i)2-s + (−0.288 − 0.499i)3-s + (0.324 − 0.562i)4-s − 0.315i·5-s + (0.642 + 0.370i)6-s + (0.513 + 0.296i)7-s − 0.450i·8-s + (−0.166 + 0.288i)9-s + (0.202 + 0.351i)10-s + (−0.261 + 0.150i)11-s − 0.375·12-s + (−0.303 − 0.952i)13-s − 0.761·14-s + (−0.157 + 0.0911i)15-s + (0.613 + 1.06i)16-s + (−0.403 + 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397631 - 0.294642i\)
\(L(\frac12)\) \(\approx\) \(0.397631 - 0.294642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (1.09 + 3.43i)T \)
good2 \( 1 + (1.57 - 0.908i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.705iT - 5T^{2} \)
7 \( 1 + (-1.35 - 0.784i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (1.66 - 2.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.162 + 0.0938i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.31 + 7.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.38 + 7.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (-5.32 + 3.07i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.60 + 0.928i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.515 + 0.893i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 - 0.146T + 53T^{2} \)
59 \( 1 + (-8.64 - 4.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.79 - 6.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.07 + 2.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.92 + 3.42i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 - 16.0iT - 83T^{2} \)
89 \( 1 + (-6.61 + 3.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.35 + 0.784i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69095541063424877073543762380, −10.02118679817901048110588808954, −8.849957854765497696262642509742, −8.182777833169136434958805203246, −7.56994887348034211244797173996, −6.46235886702071707277976733871, −5.58542123375534455400523354722, −4.22678202380712051742572142007, −2.22957196957648181843817496543, −0.48658128006752418731807951665, 1.52124690561893691764780168631, 2.98559229978247683297443612648, 4.49839354962126072149076471410, 5.51463543152287976795398135466, 6.96792453838392114082872223626, 7.88484407157410270943121674202, 8.981969310905021259346711038964, 9.582375972442713856142814312741, 10.42501111297035637054838530845, 11.33821601389640254683125101727

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