Properties

Label 2-6-1.1-c25-0-2
Degree $2$
Conductor $6$
Sign $-1$
Analytic cond. $23.7598$
Root an. cond. $4.87440$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3·2-s − 5.31e5·3-s + 1.67e7·4-s − 2.92e8·5-s + 2.17e9·6-s + 3.58e9·7-s − 6.87e10·8-s + 2.82e11·9-s + 1.19e12·10-s + 1.51e13·11-s − 8.91e12·12-s + 1.22e12·13-s − 1.46e13·14-s + 1.55e14·15-s + 2.81e14·16-s + 2.51e15·17-s − 1.15e15·18-s − 7.99e15·19-s − 4.91e15·20-s − 1.90e15·21-s − 6.18e16·22-s − 9.96e16·23-s + 3.65e16·24-s − 2.12e17·25-s − 5.00e15·26-s − 1.50e17·27-s + 6.00e16·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.536·5-s + 0.408·6-s + 0.0977·7-s − 0.353·8-s + 1/3·9-s + 0.379·10-s + 1.45·11-s − 0.288·12-s + 0.0145·13-s − 0.0691·14-s + 0.309·15-s + 1/4·16-s + 1.04·17-s − 0.235·18-s − 0.828·19-s − 0.268·20-s − 0.0564·21-s − 1.02·22-s − 0.948·23-s + 0.204·24-s − 0.712·25-s − 0.0102·26-s − 0.192·27-s + 0.0488·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-1$
Analytic conductor: \(23.7598\)
Root analytic conductor: \(4.87440\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6,\ (\ :25/2),\ -1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{12} T \)
3 \( 1 + p^{12} T \)
good5 \( 1 + 11710194 p^{2} T + p^{25} T^{2} \)
7 \( 1 - 73074368 p^{2} T + p^{25} T^{2} \)
11 \( 1 - 1373779385292 p T + p^{25} T^{2} \)
13 \( 1 - 1221071681246 T + p^{25} T^{2} \)
17 \( 1 - 2518250853863682 T + p^{25} T^{2} \)
19 \( 1 + 7992693407413060 T + p^{25} T^{2} \)
23 \( 1 + 99645642629247624 T + p^{25} T^{2} \)
29 \( 1 + 2080672742244316890 T + p^{25} T^{2} \)
31 \( 1 + 4937672075835729208 T + p^{25} T^{2} \)
37 \( 1 - 19829154107621718182 T + p^{25} T^{2} \)
41 \( 1 - \)\(22\!\cdots\!42\)\( T + p^{25} T^{2} \)
43 \( 1 + 72221008334482349884 T + p^{25} T^{2} \)
47 \( 1 - \)\(18\!\cdots\!92\)\( T + p^{25} T^{2} \)
53 \( 1 + \)\(26\!\cdots\!74\)\( T + p^{25} T^{2} \)
59 \( 1 + \)\(16\!\cdots\!40\)\( T + p^{25} T^{2} \)
61 \( 1 + \)\(35\!\cdots\!38\)\( T + p^{25} T^{2} \)
67 \( 1 - \)\(10\!\cdots\!92\)\( T + p^{25} T^{2} \)
71 \( 1 - \)\(73\!\cdots\!12\)\( T + p^{25} T^{2} \)
73 \( 1 + \)\(26\!\cdots\!74\)\( T + p^{25} T^{2} \)
79 \( 1 + \)\(10\!\cdots\!40\)\( T + p^{25} T^{2} \)
83 \( 1 - \)\(15\!\cdots\!96\)\( T + p^{25} T^{2} \)
89 \( 1 - \)\(21\!\cdots\!90\)\( T + p^{25} T^{2} \)
97 \( 1 + \)\(44\!\cdots\!58\)\( T + p^{25} T^{2} \)
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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

−16.28390179013598741891130728815, −14.65124013432803526774324492695, −12.27474619047418101939762531155, −11.13311874899365858884242757943, −9.447069009122651123552412829247, −7.69675648447746439842264778607, −6.10369125511704443542961965420, −3.91156177920039508566881151426, −1.52925664068154823224782772760, 0, 1.52925664068154823224782772760, 3.91156177920039508566881151426, 6.10369125511704443542961965420, 7.69675648447746439842264778607, 9.447069009122651123552412829247, 11.13311874899365858884242757943, 12.27474619047418101939762531155, 14.65124013432803526774324492695, 16.28390179013598741891130728815

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