Properties

Label 2-429-429.311-c0-0-2
Degree $2$
Conductor $429$
Sign $0.794 + 0.606i$
Analytic cond. $0.214098$
Root an. cond. $0.462708$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.5 − 0.363i)5-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (0.190 − 0.587i)15-s + (−0.499 + 1.53i)18-s + (−0.809 + 0.587i)20-s − 1.61·22-s + ⋯
L(s)  = 1  + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.5 − 0.363i)5-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (0.190 − 0.587i)15-s + (−0.499 + 1.53i)18-s + (−0.809 + 0.587i)20-s − 1.61·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526638123\)
\(L(\frac12)\) \(\approx\) \(1.526638123\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−11.49664824254612152427935132512, −10.56831580948536694838757312909, −9.889951305976609608343033165024, −8.716194392510235710063036383830, −7.75808725216106026852526261124, −6.03462950567384111738172190494, −4.98623120538750706106171505842, −4.40620994771257051691464826572, −3.36516045675703410611326529525, −2.38355047805118523514237716335, 2.53536426066171617188298728752, 3.58722485375291194822162730118, 4.91429579640511628840861985634, 5.81474420226351671011310507734, 6.88297696684549833608361133753, 7.54582421722621227070330404234, 8.040127292498926733022239963097, 9.535548659844534397633048761481, 10.88548153941724394379853471341, 11.97963818612987967760176749008

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