Properties

Label 2-6-1.1-c15-0-2
Degree $2$
Conductor $6$
Sign $-1$
Analytic cond. $8.56161$
Root an. cond. $2.92602$
Motivic weight $15$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128·2-s − 2.18e3·3-s + 1.63e4·4-s − 1.14e5·5-s − 2.79e5·6-s − 3.03e6·7-s + 2.09e6·8-s + 4.78e6·9-s − 1.46e7·10-s − 1.03e8·11-s − 3.58e7·12-s − 1.04e8·13-s − 3.88e8·14-s + 2.51e8·15-s + 2.68e8·16-s + 9.97e8·17-s + 6.12e8·18-s + 4.93e9·19-s − 1.88e9·20-s + 6.63e9·21-s − 1.32e10·22-s + 8.32e9·23-s − 4.58e9·24-s − 1.73e10·25-s − 1.33e10·26-s − 1.04e10·27-s − 4.97e10·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.657·5-s − 0.408·6-s − 1.39·7-s + 0.353·8-s + 1/3·9-s − 0.464·10-s − 1.60·11-s − 0.288·12-s − 0.461·13-s − 0.984·14-s + 0.379·15-s + 1/4·16-s + 0.589·17-s + 0.235·18-s + 1.26·19-s − 0.328·20-s + 0.804·21-s − 1.13·22-s + 0.509·23-s − 0.204·24-s − 0.568·25-s − 0.326·26-s − 0.192·27-s − 0.696·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-1$
Analytic conductor: \(8.56161\)
Root analytic conductor: \(2.92602\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6,\ (\ :15/2),\ -1)\)

Particular Values

\(L(8)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{7} T \)
3 \( 1 + p^{7} T \)
good5 \( 1 + 22962 p T + p^{15} T^{2} \)
7 \( 1 + 433504 p T + p^{15} T^{2} \)
11 \( 1 + 9404700 p T + p^{15} T^{2} \)
13 \( 1 + 104365834 T + p^{15} T^{2} \)
17 \( 1 - 997689762 T + p^{15} T^{2} \)
19 \( 1 - 4934015444 T + p^{15} T^{2} \)
23 \( 1 - 8324920200 T + p^{15} T^{2} \)
29 \( 1 - 104128242846 T + p^{15} T^{2} \)
31 \( 1 + 296696681512 T + p^{15} T^{2} \)
37 \( 1 + 178337455666 T + p^{15} T^{2} \)
41 \( 1 + 1790882416086 T + p^{15} T^{2} \)
43 \( 1 + 2863459422772 T + p^{15} T^{2} \)
47 \( 1 - 4332907521600 T + p^{15} T^{2} \)
53 \( 1 - 9732317104422 T + p^{15} T^{2} \)
59 \( 1 + 13514837176500 T + p^{15} T^{2} \)
61 \( 1 - 5352663511190 T + p^{15} T^{2} \)
67 \( 1 + 53233909720108 T + p^{15} T^{2} \)
71 \( 1 + 20229661643400 T + p^{15} T^{2} \)
73 \( 1 - 26264166466106 T + p^{15} T^{2} \)
79 \( 1 + 339031361615128 T + p^{15} T^{2} \)
83 \( 1 - 131684771045076 T + p^{15} T^{2} \)
89 \( 1 + 39352148322678 T + p^{15} T^{2} \)
97 \( 1 - 1128750908801474 T + p^{15} T^{2} \)
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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

−18.59638828057091361754563196311, −16.40622369628198614724642650675, −15.50932234651990188693513339586, −13.27043991391127981254467363707, −12.04994408283814145233740656932, −10.21600059651067033387315001498, −7.28892296803187118639224467436, −5.37856479530743569026031590586, −3.21748929729541942268506001835, 0, 3.21748929729541942268506001835, 5.37856479530743569026031590586, 7.28892296803187118639224467436, 10.21600059651067033387315001498, 12.04994408283814145233740656932, 13.27043991391127981254467363707, 15.50932234651990188693513339586, 16.40622369628198614724642650675, 18.59638828057091361754563196311

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