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Properties

Label 2-429-13.12-c1-0-10
Degree $2$
Conductor $429$
Sign $0.966 + 0.255i$
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  − 1.36i·2-s + 3-s + 0.128·4-s + 2.18i·5-s − 1.36i·6-s + 4.27i·7-s − 2.91i·8-s + 9-s + 2.98·10-s − i·11-s + 0.128·12-s + (−0.922 + 3.48i)13-s + 5.84·14-s + 2.18i·15-s − 3.72·16-s + 3.79·17-s + ⋯
L(s)  = 1  − 0.967i·2-s + 0.577·3-s + 0.0640·4-s + 0.976i·5-s − 0.558i·6-s + 1.61i·7-s − 1.02i·8-s + 0.333·9-s + 0.944·10-s − 0.301i·11-s + 0.0370·12-s + (−0.255 + 0.966i)13-s + 1.56·14-s + 0.563i·15-s − 0.931·16-s + 0.919·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.966 + 0.255i$
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: \((2,\ 429,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85525 - 0.241341i\)
\(L(\frac12)\) \(\approx\) \(1.85525 - 0.241341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.922 - 3.48i)T \)
good2 \( 1 + 1.36iT - 2T^{2} \)
5 \( 1 - 2.18iT - 5T^{2} \)
7 \( 1 - 4.27iT - 7T^{2} \)
17 \( 1 - 3.79T + 17T^{2} \)
19 \( 1 + 3.17iT - 19T^{2} \)
23 \( 1 - 4.94T + 23T^{2} \)
29 \( 1 + 7.35T + 29T^{2} \)
31 \( 1 - 0.0727iT - 31T^{2} \)
37 \( 1 + 3.66iT - 37T^{2} \)
41 \( 1 + 4.16iT - 41T^{2} \)
43 \( 1 + 7.11T + 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 - 5.11T + 53T^{2} \)
59 \( 1 + 5.62iT - 59T^{2} \)
61 \( 1 + 5.40T + 61T^{2} \)
67 \( 1 - 10.1iT - 67T^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 + 6.44iT - 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 7.80iT - 83T^{2} \)
89 \( 1 + 2.87iT - 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−11.28819782197258415367476720139, −10.32477811306648995158889759658, −9.366321098842332248136258447738, −8.783960945478763714355198237363, −7.33413091935819557464902887654, −6.57755472582170690130764530938, −5.32209884418142877255568229670, −3.61943689996027661652359597756, −2.77427391614251054057964748137, −1.98628983682244623158151723082, 1.30153165306499893560337598840, 3.26024751275536021365570087787, 4.56522898148011419524865355262, 5.46235661108231291887080850284, 6.76281216606276824147198774015, 7.72706396401348546409622358645, 7.976920086131455179159461176841, 9.223776444078252151474443324699, 10.19011643836144932740455857594, 11.02724820522840545272226348797

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