Properties

Label 2-429-13.12-c1-0-3
Degree $2$
Conductor $429$
Sign $-0.811 + 0.584i$
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  + 2.53i·2-s + 3-s − 4.42·4-s + 3.70i·5-s + 2.53i·6-s + 0.957i·7-s − 6.14i·8-s + 9-s − 9.37·10-s i·11-s − 4.42·12-s + (−2.10 − 2.92i)13-s − 2.42·14-s + 3.70i·15-s + 6.71·16-s + 2.05·17-s + ⋯
L(s)  = 1  + 1.79i·2-s + 0.577·3-s − 2.21·4-s + 1.65i·5-s + 1.03i·6-s + 0.361i·7-s − 2.17i·8-s + 0.333·9-s − 2.96·10-s − 0.301i·11-s − 1.27·12-s + (−0.584 − 0.811i)13-s − 0.648·14-s + 0.955i·15-s + 1.67·16-s + 0.498·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.811 + 0.584i$
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: \((2,\ 429,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412402 - 1.27721i\)
\(L(\frac12)\) \(\approx\) \(0.412402 - 1.27721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 2.92i)T \)
good2 \( 1 - 2.53iT - 2T^{2} \)
5 \( 1 - 3.70iT - 5T^{2} \)
7 \( 1 - 0.957iT - 7T^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 - 7.67iT - 19T^{2} \)
23 \( 1 - 4.20T + 23T^{2} \)
29 \( 1 - 1.97T + 29T^{2} \)
31 \( 1 + 10.5iT - 31T^{2} \)
37 \( 1 - 8.30iT - 37T^{2} \)
41 \( 1 + 5.23iT - 41T^{2} \)
43 \( 1 + 5.26T + 43T^{2} \)
47 \( 1 - 1.24iT - 47T^{2} \)
53 \( 1 - 2.98T + 53T^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 + 0.183T + 61T^{2} \)
67 \( 1 - 1.40iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 - 3.32iT - 73T^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 - 3.31iT - 83T^{2} \)
89 \( 1 + 5.24iT - 89T^{2} \)
97 \( 1 - 9.82iT - 97T^{2} \)
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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

−11.74605142691315489262872537624, −10.33461611081843531831457049198, −9.809046244561676553086416724944, −8.551303610142100012398402353964, −7.76594331083568893535627469292, −7.19656930841188287157064841860, −6.19503196693959207976934850392, −5.49742873221359328325766117892, −3.92280503618039896378287978775, −2.80288778278513741670823146783, 0.854368405444295335915957851674, 2.02670089233783380950797050614, 3.36548166472854992964484530046, 4.66612251210297574812257396192, 4.90257933597137854145576069667, 7.14658176880682044203257835802, 8.525954977268712685278368872116, 9.086416555332596233455562421498, 9.630458091740432491965958019528, 10.65860975853026005743225035914

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