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Properties

Label 6-2175e3-87.86-c0e3-0-1
Degree 66
Conductor 1028910937510289109375
Sign 11
Analytic cond. 1.278931.27893
Root an. cond. 1.041851.04185
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 8-s + 6·9-s − 3·24-s − 10·27-s + 3·29-s − 3·41-s + 6·72-s + 15·81-s − 9·87-s + 3·103-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3·3-s + 8-s + 6·9-s − 3·24-s − 10·27-s + 3·29-s − 3·41-s + 6·72-s + 15·81-s − 9·87-s + 3·103-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅293)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((33⋅56⋅293)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅2933^{3} \cdot 5^{6} \cdot 29^{3}
Sign: 11
Analytic conductor: 1.278931.27893
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ2175(1826,⋅)\chi_{2175} (1826, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅293, ( :0,0,0), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 29^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52290481400.5229048140
L(12)L(\frac12) ≈\approx 0.52290481400.5229048140
L(1)L(1) 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+T)3 ( 1 + T )^{3}
5 1 1
29C1C_1 (1−T)3 ( 1 - T )^{3}
good2C6C_6 1−T3+T6 1 - T^{3} + T^{6}
7C6C_6 1−T3+T6 1 - T^{3} + T^{6}
11C6C_6 1+T3+T6 1 + T^{3} + T^{6}
13C6C_6 1−T3+T6 1 - T^{3} + T^{6}
17C6C_6 1−T3+T6 1 - T^{3} + T^{6}
19C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
23C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
31C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
37C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
41C2C_2 (1+T+T2)3 ( 1 + T + T^{2} )^{3}
43C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
47C6C_6 1−T3+T6 1 - T^{3} + T^{6}
53C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
59C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
61C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
67C6C_6 1−T3+T6 1 - T^{3} + T^{6}
71C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
73C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
79C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
83C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
89C6C_6 1+T3+T6 1 + T^{3} + T^{6}
97C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.292539683389288985112163405555, −7.71621506664115351524724123027, −7.68448898919329146098893357109, −7.28625155324289672698272817718, −6.99658498534019354268492847128, −6.91906838436844395268577660582, −6.57361497476501462714389800745, −6.33749050401458408994139787064, −6.07897706859304921688609895872, −6.06636784162407590066444391644, −5.37791348032863760307696527283, −5.26844723531087514107282166394, −5.05295758961349030166920206413, −4.83731914147215123808324809836, −4.53080275361845858840565894443, −4.35136885190179736969885903481, −4.02100546134468548233210777415, −3.69950209953707938992029285562, −3.13302973908668216888296534943, −2.98179984216700250557221153793, −2.06303999472383132311104479016, −1.93003054497361431391125367017, −1.47579939597712814270817544533, −1.05380231752988230136180855778, −0.62600961268415776705364245821, 0.62600961268415776705364245821, 1.05380231752988230136180855778, 1.47579939597712814270817544533, 1.93003054497361431391125367017, 2.06303999472383132311104479016, 2.98179984216700250557221153793, 3.13302973908668216888296534943, 3.69950209953707938992029285562, 4.02100546134468548233210777415, 4.35136885190179736969885903481, 4.53080275361845858840565894443, 4.83731914147215123808324809836, 5.05295758961349030166920206413, 5.26844723531087514107282166394, 5.37791348032863760307696527283, 6.06636784162407590066444391644, 6.07897706859304921688609895872, 6.33749050401458408994139787064, 6.57361497476501462714389800745, 6.91906838436844395268577660582, 6.99658498534019354268492847128, 7.28625155324289672698272817718, 7.68448898919329146098893357109, 7.71621506664115351524724123027, 8.292539683389288985112163405555

Graph of the ZZ-function along the critical line