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Properties

Label 2-12-1.1-c3-0-0
Degree 22
Conductor 1212
Sign 11
Analytic cond. 0.7080220.708022
Root an. cond. 0.8414400.841440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 22
Conductor: 1212    =    22⋅32^{2} \cdot 3
Sign: 11
Analytic conductor: 0.7080220.708022
Root analytic conductor: 0.8414400.841440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 12, ( :3/2), 1)(2,\ 12,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.93444013810.9344401381
L(12)L(\frac12) ≈\approx 0.93444013810.9344401381
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−pT 1 - p T
good5 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
7 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
13 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
17 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
19 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
23 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
29 1+234T+p3T2 1 + 234 T + p^{3} T^{2}
31 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
37 1+226T+p3T2 1 + 226 T + p^{3} T^{2}
41 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
43 1−452T+p3T2 1 - 452 T + p^{3} T^{2}
47 1−432T+p3T2 1 - 432 T + p^{3} T^{2}
53 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
59 1+684T+p3T2 1 + 684 T + p^{3} T^{2}
61 1−422T+p3T2 1 - 422 T + p^{3} T^{2}
67 1−332T+p3T2 1 - 332 T + p^{3} T^{2}
71 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
73 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
79 1−512T+p3T2 1 - 512 T + p^{3} T^{2}
83 1+1188T+p3T2 1 + 1188 T + p^{3} T^{2}
89 1+630T+p3T2 1 + 630 T + p^{3} T^{2}
97 1+1054T+p3T2 1 + 1054 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line