Properties

Label 2-429-13.12-c1-0-11
Degree $2$
Conductor $429$
Sign $0.246 - 0.969i$
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  + 0.584i·2-s + 3-s + 1.65·4-s + 1.95i·5-s + 0.584i·6-s + 1.51i·7-s + 2.13i·8-s + 9-s − 1.14·10-s + i·11-s + 1.65·12-s + (−3.49 − 0.887i)13-s − 0.883·14-s + 1.95i·15-s + 2.06·16-s − 3.44·17-s + ⋯
L(s)  = 1  + 0.413i·2-s + 0.577·3-s + 0.829·4-s + 0.873i·5-s + 0.238i·6-s + 0.571i·7-s + 0.756i·8-s + 0.333·9-s − 0.361·10-s + 0.301i·11-s + 0.478·12-s + (−0.969 − 0.246i)13-s − 0.236·14-s + 0.504i·15-s + 0.516·16-s − 0.834·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.246 - 0.969i$
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: \((2,\ 429,\ (\ :1/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54348 + 1.20057i\)
\(L(\frac12)\) \(\approx\) \(1.54348 + 1.20057i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.49 + 0.887i)T \)
good2 \( 1 - 0.584iT - 2T^{2} \)
5 \( 1 - 1.95iT - 5T^{2} \)
7 \( 1 - 1.51iT - 7T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 + 5.09iT - 19T^{2} \)
23 \( 1 - 0.701T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 + 7.26iT - 31T^{2} \)
37 \( 1 + 2.08iT - 37T^{2} \)
41 \( 1 + 1.03iT - 41T^{2} \)
43 \( 1 - 5.48T + 43T^{2} \)
47 \( 1 - 3.75iT - 47T^{2} \)
53 \( 1 + 0.502T + 53T^{2} \)
59 \( 1 + 2.65iT - 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 - 8.47iT - 67T^{2} \)
71 \( 1 - 2.31iT - 71T^{2} \)
73 \( 1 - 2.21iT - 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 + 1.80iT - 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−11.26710015653964920190889099012, −10.53049200547432708271435077679, −9.497065591949274685774131950980, −8.513894325224058505171232152240, −7.40281376099407010356833071325, −6.92596601527640565959811202953, −5.89730999452293035802371657272, −4.59388671215904527114519050901, −2.84064756745205897963106416112, −2.34953874392678266094555662051, 1.30745626498582484053715845528, 2.62580869438595163665181359938, 3.87817015711087439486930593427, 4.97995114603887845004452581370, 6.43666889343930551478988413221, 7.32477649566249607091491712694, 8.269059945831322955959436217463, 9.213018668749708902588414532791, 10.18491770414577745398362562435, 10.85037646187942047489180082175

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