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Properties

Label 2-429-13.10-c1-0-6
Degree $2$
Conductor $429$
Sign $0.615 - 0.788i$
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  + (−0.489 + 0.282i)2-s + (−0.5 − 0.866i)3-s + (−0.840 + 1.45i)4-s − 1.14i·5-s + (0.489 + 0.282i)6-s + (2.75 + 1.59i)7-s − 2.08i·8-s + (−0.499 + 0.866i)9-s + (0.323 + 0.560i)10-s + (−0.866 + 0.5i)11-s + 1.68·12-s + (−2.18 + 2.87i)13-s − 1.79·14-s + (−0.991 + 0.572i)15-s + (−1.09 − 1.89i)16-s + (1.58 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.346 + 0.199i)2-s + (−0.288 − 0.499i)3-s + (−0.420 + 0.727i)4-s − 0.512i·5-s + (0.199 + 0.115i)6-s + (1.04 + 0.601i)7-s − 0.735i·8-s + (−0.166 + 0.288i)9-s + (0.102 + 0.177i)10-s + (−0.261 + 0.150i)11-s + 0.485·12-s + (−0.604 + 0.796i)13-s − 0.480·14-s + (−0.256 + 0.147i)15-s + (−0.272 − 0.472i)16-s + (0.384 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.615 - 0.788i$
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: \((2,\ 429,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890712 + 0.434859i\)
\(L(\frac12)\) \(\approx\) \(0.890712 + 0.434859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 2.87i)T \)
good2 \( 1 + (0.489 - 0.282i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.14iT - 5T^{2} \)
7 \( 1 + (-2.75 - 1.59i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-1.58 + 2.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.59 - 2.65i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.16 - 5.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.72 - 6.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.39iT - 31T^{2} \)
37 \( 1 + (5.06 - 2.92i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.55 + 1.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.14 + 5.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.76iT - 47T^{2} \)
53 \( 1 + 0.509T + 53T^{2} \)
59 \( 1 + (1.29 + 0.746i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.23 - 7.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.40 + 1.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.96 + 4.02i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.446iT - 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + (-7.51 + 4.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.73 - 1.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.72865839174841011231092790456, −10.32055458479743778086356908952, −9.100171348019178185296996104221, −8.647509463755054167496163392414, −7.55047874801957584040930422832, −7.04794539796433335381095950706, −5.29775346485237297222099579725, −4.81806302336798232158064939011, −3.13901385837397222719303937183, −1.42852562826158657179989482877, 0.863340996792215245428014289518, 2.70744350137565747503932862258, 4.38917442650046122037321825375, 5.12411644295090468032058038928, 6.13971240599791579883411930963, 7.51864443544242248494676215034, 8.343678140726086658904615378423, 9.487145482266810828064028271946, 10.28345470823686357584234963657, 10.85525923551307593474773190469

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