Properties

Label 4-329e2-1.1-c0e2-0-0
Degree 44
Conductor 108241108241
Sign 11
Analytic cond. 0.02695910.0269591
Root an. cond. 0.4052060.405206
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯
L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯

Functional equation

Λ(s)=(108241s/2ΓC(s)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 108241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(108241s/2ΓC(s)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 108241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 108241108241    =    724727^{2} \cdot 47^{2}
Sign: 11
Analytic conductor: 0.02695910.0269591
Root analytic conductor: 0.4052060.405206
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 108241, ( :0,0), 1)(4,\ 108241,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.59820805860.5982080586
L(12)L(\frac12) \approx 0.59820805860.5982080586
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad7C2C_2 1+T+T2 1 + T + T^{2}
47C2C_2 1+T+T2 1 + T + T^{2}
good2C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
3C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
5C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
11C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
17C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
19C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
23C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
41C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
59C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
61C1C_1×\timesC2C_2 (1T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
73C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
83C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
89C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
97C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.83988006959751153266928827298, −11.61522390432652797060404954170, −11.34400716457985881643436985485, −10.90672704503313158336709174108, −10.16755714907701894495099559863, −10.16158649940227520378359334706, −9.688275521645396995568294994764, −8.624463388835655697389509257478, −8.169032658583608566069655164604, −7.46253857929727232494147872029, −6.95398221082477526294204008166, −6.62079931918439691961234634925, −5.92795201129495718593740288338, −5.63318403982791156057331750624, −5.40786190130025409057174137241, −4.54907603163398575805880359415, −4.19381485711651291004855249184, −3.29721858350856258410418911185, −2.63006638421219917034747835911, −1.31151289050695141982660107552, 1.31151289050695141982660107552, 2.63006638421219917034747835911, 3.29721858350856258410418911185, 4.19381485711651291004855249184, 4.54907603163398575805880359415, 5.40786190130025409057174137241, 5.63318403982791156057331750624, 5.92795201129495718593740288338, 6.62079931918439691961234634925, 6.95398221082477526294204008166, 7.46253857929727232494147872029, 8.169032658583608566069655164604, 8.624463388835655697389509257478, 9.688275521645396995568294994764, 10.16158649940227520378359334706, 10.16755714907701894495099559863, 10.90672704503313158336709174108, 11.34400716457985881643436985485, 11.61522390432652797060404954170, 11.83988006959751153266928827298

Graph of the ZZ-function along the critical line