Properties

Label 2-12-1.1-c3-0-0
Degree $2$
Conductor $12$
Sign $1$
Analytic cond. $0.708022$
Root an. cond. $0.841440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 18·5-s + 8·7-s + 9·9-s + 36·11-s − 10·13-s − 54·15-s + 18·17-s − 100·19-s + 24·21-s + 72·23-s + 199·25-s + 27·27-s − 234·29-s − 16·31-s + 108·33-s − 144·35-s − 226·37-s − 30·39-s + 90·41-s + 452·43-s − 162·45-s + 432·47-s − 279·49-s + 54·51-s + 414·53-s − 648·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.60·5-s + 0.431·7-s + 1/3·9-s + 0.986·11-s − 0.213·13-s − 0.929·15-s + 0.256·17-s − 1.20·19-s + 0.249·21-s + 0.652·23-s + 1.59·25-s + 0.192·27-s − 1.49·29-s − 0.0926·31-s + 0.569·33-s − 0.695·35-s − 1.00·37-s − 0.123·39-s + 0.342·41-s + 1.60·43-s − 0.536·45-s + 1.34·47-s − 0.813·49-s + 0.148·51-s + 1.07·53-s − 1.58·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.708022\)
Root analytic conductor: \(0.841440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9344401381\)
\(L(\frac12)\) \(\approx\) \(0.9344401381\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
good5 \( 1 + 18 T + p^{3} T^{2} \)
7 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 + 10 T + p^{3} T^{2} \)
17 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 + 234 T + p^{3} T^{2} \)
31 \( 1 + 16 T + p^{3} T^{2} \)
37 \( 1 + 226 T + p^{3} T^{2} \)
41 \( 1 - 90 T + p^{3} T^{2} \)
43 \( 1 - 452 T + p^{3} T^{2} \)
47 \( 1 - 432 T + p^{3} T^{2} \)
53 \( 1 - 414 T + p^{3} T^{2} \)
59 \( 1 + 684 T + p^{3} T^{2} \)
61 \( 1 - 422 T + p^{3} T^{2} \)
67 \( 1 - 332 T + p^{3} T^{2} \)
71 \( 1 + 360 T + p^{3} T^{2} \)
73 \( 1 - 26 T + p^{3} T^{2} \)
79 \( 1 - 512 T + p^{3} T^{2} \)
83 \( 1 + 1188 T + p^{3} T^{2} \)
89 \( 1 + 630 T + p^{3} T^{2} \)
97 \( 1 + 1054 T + p^{3} T^{2} \)
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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

−19.65697221873960303781526402990, −18.91719553273081271326643124779, −16.94656258097691856593765242257, −15.42632448273370742373967743432, −14.49301964336732569099039232284, −12.46785072595897989496432329095, −11.13009747514994459636212243893, −8.800395059796186281786333535307, −7.38784077424888441275854326091, −4.02436353376270990090525228298, 4.02436353376270990090525228298, 7.38784077424888441275854326091, 8.800395059796186281786333535307, 11.13009747514994459636212243893, 12.46785072595897989496432329095, 14.49301964336732569099039232284, 15.42632448273370742373967743432, 16.94656258097691856593765242257, 18.91719553273081271326643124779, 19.65697221873960303781526402990

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