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Properties

Label 2-429-13.4-c1-0-23
Degree $2$
Conductor $429$
Sign $-0.122 + 0.992i$
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  + (0.962 + 0.555i)2-s + (−0.5 + 0.866i)3-s + (−0.381 − 0.661i)4-s − 2.79i·5-s + (−0.962 + 0.555i)6-s + (−3.71 + 2.14i)7-s − 3.07i·8-s + (−0.499 − 0.866i)9-s + (1.55 − 2.68i)10-s + (−0.866 − 0.5i)11-s + 0.763·12-s + (0.486 − 3.57i)13-s − 4.76·14-s + (2.41 + 1.39i)15-s + (0.944 − 1.63i)16-s + (−1.73 − 3.00i)17-s + ⋯
L(s)  = 1  + (0.680 + 0.393i)2-s + (−0.288 + 0.499i)3-s + (−0.190 − 0.330i)4-s − 1.24i·5-s + (−0.393 + 0.226i)6-s + (−1.40 + 0.810i)7-s − 1.08i·8-s + (−0.166 − 0.288i)9-s + (0.490 − 0.849i)10-s + (−0.261 − 0.150i)11-s + 0.220·12-s + (0.134 − 0.990i)13-s − 1.27·14-s + (0.624 + 0.360i)15-s + (0.236 − 0.408i)16-s + (−0.420 − 0.728i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630192 - 0.712598i\)
\(L(\frac12)\) \(\approx\) \(0.630192 - 0.712598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.486 + 3.57i)T \)
good2 \( 1 + (-0.962 - 0.555i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.79iT - 5T^{2} \)
7 \( 1 + (3.71 - 2.14i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (1.73 + 3.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.26 - 1.31i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.38 + 4.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.784 - 1.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.371iT - 31T^{2} \)
37 \( 1 + (0.0762 + 0.0440i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.94 - 2.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.26 + 3.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 7.17T + 53T^{2} \)
59 \( 1 + (-5.52 + 3.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.41 + 12.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.8 - 7.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.143 + 0.0825i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.12iT - 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + 0.0251iT - 83T^{2} \)
89 \( 1 + (6.34 + 3.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.44 + 3.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.81873355950013010033677450634, −9.830812179770229195338750938243, −9.226382911274069319710537795411, −8.430278354439010134362646244844, −6.76232978725952765773020003952, −5.84568403555903184356519503018, −5.24069749140152248298269030928, −4.31105434209772064534368788263, −3.02081251801334587974012410864, −0.48853588378325215199334449994, 2.35676133257149353920775267562, 3.41473785214077806892245043927, 4.23679964512374792000526205790, 5.84507292395346375449480361066, 6.82131587021211942027361387082, 7.28836319901293671946637408931, 8.671839196731266141957664535176, 9.863566473175851608607909004606, 10.78239054113177755543781156685, 11.40489667602099880147077841721

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