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Properties

Label 2-6336-12.11-c1-0-54
Degree $2$
Conductor $6336$
Sign $0.577 + 0.816i$
Analytic cond. $50.5932$
Root an. cond. $7.11289$
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  + 0.558i·5-s − 0.558i·7-s + 11-s + 1.30·13-s + 5.94i·17-s − 6.72i·19-s − 4.99·23-s + 4.68·25-s − 3.46i·29-s − 2.48i·31-s + 0.311·35-s − 9.62·37-s − 2.48i·41-s − 2.87i·43-s + 6.92·47-s + ⋯
L(s)  = 1  + 0.249i·5-s − 0.210i·7-s + 0.301·11-s + 0.362·13-s + 1.44i·17-s − 1.54i·19-s − 1.04·23-s + 0.937·25-s − 0.643i·29-s − 0.446i·31-s + 0.0526·35-s − 1.58·37-s − 0.387i·41-s − 0.438i·43-s + 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(50.5932\)
Root analytic conductor: \(7.11289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6336} (3455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6336,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.729760100\)
\(L(\frac12)\) \(\approx\) \(1.729760100\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 0.558iT - 5T^{2} \)
7 \( 1 + 0.558iT - 7T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 - 5.94iT - 17T^{2} \)
19 \( 1 + 6.72iT - 19T^{2} \)
23 \( 1 + 4.99T + 23T^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + 2.48iT - 31T^{2} \)
37 \( 1 + 9.62T + 37T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + 2.87iT - 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 7.48iT - 53T^{2} \)
59 \( 1 - 3.32T + 59T^{2} \)
61 \( 1 + 2.69T + 61T^{2} \)
67 \( 1 + 5.02iT - 67T^{2} \)
71 \( 1 + 4.99T + 71T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 1.56iT - 89T^{2} \)
97 \( 1 - 3.68T + 97T^{2} \)
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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

−7.958847838755561543483727335039, −7.11150618351212732127400846298, −6.58428997210193228465676949841, −5.87832634992396059096678434021, −5.08280844222629248792203126392, −4.13207349117410665507166023703, −3.64834033222593647559342458558, −2.57781006846515289783907127659, −1.73534572111326174726991382080, −0.50385806728382670605820336777, 0.971227525078215168247536401046, 1.90989514499797520576873265290, 2.96460913214935629444419482476, 3.71639854487091061226789397613, 4.55892921341280826160585368442, 5.33928892870080472147144004591, 5.94892361934915189611532288618, 6.76791027850936517587642241377, 7.44145229883941607192988298012, 8.151978755332211635024671941754

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