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Properties

Label 2-429-13.12-c1-0-2
Degree $2$
Conductor $429$
Sign $0.868 + 0.494i$
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  − 2.73i·2-s + 3-s − 5.50·4-s + 2.84i·5-s − 2.73i·6-s + 3.93i·7-s + 9.58i·8-s + 9-s + 7.78·10-s + i·11-s − 5.50·12-s + (1.78 − 3.13i)13-s + 10.7·14-s + 2.84i·15-s + 15.2·16-s − 3.81·17-s + ⋯
L(s)  = 1  − 1.93i·2-s + 0.577·3-s − 2.75·4-s + 1.27i·5-s − 1.11i·6-s + 1.48i·7-s + 3.38i·8-s + 0.333·9-s + 2.46·10-s + 0.301i·11-s − 1.58·12-s + (0.494 − 0.868i)13-s + 2.87·14-s + 0.733i·15-s + 3.81·16-s − 0.926·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.868 + 0.494i$
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: \((2,\ 429,\ (\ :1/2),\ 0.868 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22429 - 0.324206i\)
\(L(\frac12)\) \(\approx\) \(1.22429 - 0.324206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 3.13i)T \)
good2 \( 1 + 2.73iT - 2T^{2} \)
5 \( 1 - 2.84iT - 5T^{2} \)
7 \( 1 - 3.93iT - 7T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 - 2.94iT - 19T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + 2.09T + 29T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 - 8.34iT - 37T^{2} \)
41 \( 1 + 6.35iT - 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 - 5.31iT - 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 5.38iT - 59T^{2} \)
61 \( 1 + 2.34T + 61T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 - 1.57T + 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 - 3.23iT - 89T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−10.87817680485522296084414091639, −10.54570158553261220851271756834, −9.416443589082080708187423630785, −8.833066635407926576710642382236, −7.85581366858483917012375659344, −6.15530072174532843388263929275, −4.90056948748038228861341980045, −3.46376011306901249901003359710, −2.82316294649373214623309007894, −1.93787664591795539504671941840, 0.816930096140076660963423538988, 4.14966644160086202866426712797, 4.33463317254518382267080219981, 5.63477008468815755791035569528, 6.82537802528995323812735638577, 7.44342462130003709879388409059, 8.428804840300003002292680173747, 9.012819542672670560450228875209, 9.693920792004362554811810251579, 11.09399511299559228973684074411

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