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Properties

Label 2-429-429.428-c1-0-17
Degree $2$
Conductor $429$
Sign $-0.166 - 0.986i$
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  + 1.50i·2-s + 1.73·3-s − 0.267·4-s − 1.75·5-s + 2.60i·6-s + 2.60i·8-s + 2.99·9-s − 2.63i·10-s + (3.27 − 0.551i)11-s − 0.464·12-s + 3.60i·13-s − 3.03·15-s − 4.46·16-s + 4.51i·18-s + 0.469·20-s + ⋯
L(s)  = 1  + 1.06i·2-s + 1.00·3-s − 0.133·4-s − 0.783·5-s + 1.06i·6-s + 0.922i·8-s + 0.999·9-s − 0.834i·10-s + (0.986 − 0.166i)11-s − 0.133·12-s + 0.999i·13-s − 0.783·15-s − 1.11·16-s + 1.06i·18-s + 0.105·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24625 + 1.47388i\)
\(L(\frac12)\) \(\approx\) \(1.24625 + 1.47388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
11 \( 1 + (-3.27 + 0.551i)T \)
13 \( 1 - 3.60iT \)
good2 \( 1 - 1.50iT - 2T^{2} \)
5 \( 1 + 1.75T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 0.469T + 59T^{2} \)
61 \( 1 - 7.21iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 16.1T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.59707888380954694239917184582, −10.40237854127923206205827426236, −9.008859359359441822673503147749, −8.704450620985675145701834851427, −7.50098558622496064787942484554, −7.10162377916752491422095548863, −6.01550606297127242808309416027, −4.51338293167750547674066886430, −3.61569968096864958134030206640, −2.02072878132481459457224833921, 1.30477914861469934398728760290, 2.75735579738725682876418946109, 3.61438339118209003024604374882, 4.46999776006899993532603180947, 6.38659936743536063469065290812, 7.45866623199609975976740891101, 8.226143172179367399902170921122, 9.356913752431085549411367940472, 9.987332304349415238567531184906, 11.00791785168883666743268955921

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