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Properties

Label 2-429-13.9-c1-0-11
Degree $2$
Conductor $429$
Sign $0.999 + 0.0256i$
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  + (0.5 − 0.866i)3-s + (1 + 1.73i)4-s + 2·5-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 1.99·12-s + (3.5 − 0.866i)13-s + (1 − 1.73i)15-s + (−1.99 + 3.46i)16-s + (−1 − 1.73i)17-s + (2 + 3.46i)19-s + (2 + 3.46i)20-s − 0.999·21-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.5 + 0.866i)4-s + 0.894·5-s + (−0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s + 0.577·12-s + (0.970 − 0.240i)13-s + (0.258 − 0.447i)15-s + (−0.499 + 0.866i)16-s + (−0.242 − 0.420i)17-s + (0.458 + 0.794i)19-s + (0.447 + 0.774i)20-s − 0.218·21-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.999 + 0.0256i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93258 - 0.0247841i\)
\(L(\frac12)\) \(\approx\) \(1.93258 - 0.0247841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + 7T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.30595891049469317905434889227, −10.24500098285570941431861656169, −9.276967248628663996638134695676, −8.282620266404962473121823690718, −7.53221004248768402197641140024, −6.47544618750378953581654077498, −5.77268977362549371154352940587, −3.96552228884396473062134756680, −2.93829981892094922936304038961, −1.64427360496605784138222353014, 1.63337268995724863201475421823, 2.81608133806649208677393965654, 4.40551627593841665537675363006, 5.62377635140552850755813229552, 6.21825399260149417762678334892, 7.30985308756402453078586037727, 8.949181378392791099305342860737, 9.235695679783324700987368205497, 10.45307335529825739568476806661, 10.75274073764671175040568899616

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