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Properties

Label 2-429-13.3-c1-0-1
Degree $2$
Conductor $429$
Sign $0.921 - 0.387i$
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  + (−0.901 − 1.56i)2-s + (−0.5 − 0.866i)3-s + (−0.624 + 1.08i)4-s − 2.90·5-s + (−0.901 + 1.56i)6-s + (−0.704 + 1.22i)7-s − 1.35·8-s + (−0.499 + 0.866i)9-s + (2.61 + 4.53i)10-s + (−0.5 − 0.866i)11-s + 1.24·12-s + (3.57 − 0.467i)13-s + 2.53·14-s + (1.45 + 2.51i)15-s + (2.46 + 4.27i)16-s + (−1.84 + 3.18i)17-s + ⋯
L(s)  = 1  + (−0.637 − 1.10i)2-s + (−0.288 − 0.499i)3-s + (−0.312 + 0.541i)4-s − 1.29·5-s + (−0.367 + 0.637i)6-s + (−0.266 + 0.461i)7-s − 0.478·8-s + (−0.166 + 0.288i)9-s + (0.828 + 1.43i)10-s + (−0.150 − 0.261i)11-s + 0.360·12-s + (0.991 − 0.129i)13-s + 0.678·14-s + (0.375 + 0.649i)15-s + (0.617 + 1.06i)16-s + (−0.446 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281559 + 0.0568103i\)
\(L(\frac12)\) \(\approx\) \(0.281559 + 0.0568103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.57 + 0.467i)T \)
good2 \( 1 + (0.901 + 1.56i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 + (0.704 - 1.22i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (1.84 - 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.79 - 4.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.960 - 1.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.17T + 31T^{2} \)
37 \( 1 + (0.171 + 0.296i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0680 - 0.117i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.63 - 9.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.74T + 47T^{2} \)
53 \( 1 + 6.17T + 53T^{2} \)
59 \( 1 + (-1.71 + 2.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.380 + 0.659i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.67 - 8.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.52 + 2.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.98T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-7.51 - 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.85 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22197063153523276519448777043, −10.73661036470063245583027555637, −9.403598545826940995015382885684, −8.606065047872375943424212920915, −7.83719057531436374195039415674, −6.63432013505116011960634748412, −5.54782568148892502749276803010, −3.87836541918883561700150682895, −2.94895237969554238485619379296, −1.34339870649854977034163216033, 0.25372327683073545453942575617, 3.30996564226002374000621572279, 4.28876275472981264247384154601, 5.55125180171447031749428274329, 6.76022476867563690321495409555, 7.33254699394247001455242389549, 8.340960260138039746356571525371, 8.972591766138359984084030796885, 10.07603781175577468150225109996, 11.07083134967709839829311961808

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