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Properties

Label 4-1769472-1.1-c1e2-0-12
Degree $4$
Conductor $1769472$
Sign $1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s + 9·11-s + 7·17-s − 9·25-s + 5·27-s − 9·33-s + 9·41-s − 6·43-s + 8·49-s − 7·51-s − 15·59-s − 8·73-s + 9·75-s + 81-s − 6·83-s + 5·89-s + 10·97-s − 18·99-s + 12·107-s − 17·113-s + 39·121-s − 9·123-s + 127-s + 6·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 2.71·11-s + 1.69·17-s − 9/5·25-s + 0.962·27-s − 1.56·33-s + 1.40·41-s − 0.914·43-s + 8/7·49-s − 0.980·51-s − 1.95·59-s − 0.936·73-s + 1.03·75-s + 1/9·81-s − 0.658·83-s + 0.529·89-s + 1.01·97-s − 1.80·99-s + 1.16·107-s − 1.59·113-s + 3.54·121-s − 0.811·123-s + 0.0887·127-s + 0.528·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.020092196\)
\(L(\frac12)\) \(\approx\) \(2.020092196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 + T + p T^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.11.aj_bq
13$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.13.a_ae
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ah_bs
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.23.a_p
29$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.29.a_an
31$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.31.a_i
37$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.37.a_aca
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.aj_bu
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.g_di
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.53.a_l
59$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.p_fc
61$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.61.a_dc
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.a_n
71$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.71.a_h
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.i_ew
79$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.79.a_k
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.g_dq
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.89.af_ha
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ak_ic
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.76816288445012016319199626469, −7.44097941417215809902168603306, −6.81544051671105572445095336501, −6.50270847059283324393831506762, −6.01213017007899066376900627948, −5.75420795026488020974932801343, −5.47480336751810664653835157533, −4.59076888942542417814186817629, −4.32623330635391613362009170443, −3.70857290744810790128431627733, −3.42519243245592785101176897219, −2.80712668167913110854208054812, −1.86902881982982207721030330322, −1.37722435516242290837260683401, −0.67220027398763645684290976825, 0.67220027398763645684290976825, 1.37722435516242290837260683401, 1.86902881982982207721030330322, 2.80712668167913110854208054812, 3.42519243245592785101176897219, 3.70857290744810790128431627733, 4.32623330635391613362009170443, 4.59076888942542417814186817629, 5.47480336751810664653835157533, 5.75420795026488020974932801343, 6.01213017007899066376900627948, 6.50270847059283324393831506762, 6.81544051671105572445095336501, 7.44097941417215809902168603306, 7.76816288445012016319199626469

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