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Properties

Label 2-429-33.32-c1-0-45
Degree $2$
Conductor $429$
Sign $-0.951 - 0.306i$
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  − 0.822·2-s + (−0.0778 − 1.73i)3-s − 1.32·4-s − 3.64i·5-s + (0.0640 + 1.42i)6-s − 1.97i·7-s + 2.73·8-s + (−2.98 + 0.269i)9-s + 2.99i·10-s + (−3.10 − 1.15i)11-s + (0.103 + 2.28i)12-s − i·13-s + 1.62i·14-s + (−6.29 + 0.283i)15-s + 0.397·16-s + 7.48·17-s + ⋯
L(s)  = 1  − 0.581·2-s + (−0.0449 − 0.998i)3-s − 0.661·4-s − 1.62i·5-s + (0.0261 + 0.581i)6-s − 0.744i·7-s + 0.966·8-s + (−0.995 + 0.0898i)9-s + 0.946i·10-s + (−0.936 − 0.349i)11-s + (0.0297 + 0.660i)12-s − 0.277i·13-s + 0.433i·14-s + (−1.62 + 0.0732i)15-s + 0.0994·16-s + 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.951 - 0.306i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0934070 + 0.593900i\)
\(L(\frac12)\) \(\approx\) \(0.0934070 + 0.593900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0778 + 1.73i)T \)
11 \( 1 + (3.10 + 1.15i)T \)
13 \( 1 + iT \)
good2 \( 1 + 0.822T + 2T^{2} \)
5 \( 1 + 3.64iT - 5T^{2} \)
7 \( 1 + 1.97iT - 7T^{2} \)
17 \( 1 - 7.48T + 17T^{2} \)
19 \( 1 - 4.73iT - 19T^{2} \)
23 \( 1 + 1.40iT - 23T^{2} \)
29 \( 1 + 0.599T + 29T^{2} \)
31 \( 1 + 3.80T + 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 + 9.36T + 41T^{2} \)
43 \( 1 - 5.55iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 7.53iT - 53T^{2} \)
59 \( 1 - 1.46iT - 59T^{2} \)
61 \( 1 + 0.106iT - 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 - 5.03iT - 71T^{2} \)
73 \( 1 - 0.797iT - 73T^{2} \)
79 \( 1 + 0.729iT - 79T^{2} \)
83 \( 1 + 1.46T + 83T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + 2.10T + 97T^{2} \)
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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

−10.40326035294665450478044808851, −9.716035477543905852498341733236, −8.486736868735071905708906987174, −8.130388175513489903381683512028, −7.41444958169747907203295370488, −5.66555598024796790490555808050, −5.08803716413122251106086198058, −3.66276602139231139023480299030, −1.47179585890061821838261081436, −0.51074082109674739371020084496, 2.62008623863648503940043648726, 3.59138880360336778798680715028, 4.97880506305577770695010519780, 5.83867964349045612234047022868, 7.26668311269894562908718146103, 8.093058366532828564234926929157, 9.238879632113973647280917340389, 9.873377105516116769068701065037, 10.55816032547054344369256944853, 11.21737370463486403889715521587

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