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Properties

Label 4-338688-1.1-c1e2-0-10
Degree $4$
Conductor $338688$
Sign $1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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 + 9-s + 8·13-s − 8·19-s − 10·25-s − 27-s − 8·31-s + 4·37-s − 8·39-s + 8·43-s − 7·49-s + 8·57-s + 8·61-s + 8·67-s + 16·73-s + 10·75-s + 8·79-s + 81-s + 8·93-s + 32·97-s + 8·103-s − 4·109-s − 4·111-s + 8·117-s + 6·121-s + 127-s − 8·129-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.21·13-s − 1.83·19-s − 2·25-s − 0.192·27-s − 1.43·31-s + 0.657·37-s − 1.28·39-s + 1.21·43-s − 49-s + 1.05·57-s + 1.02·61-s + 0.977·67-s + 1.87·73-s + 1.15·75-s + 0.900·79-s + 1/9·81-s + 0.829·93-s + 3.24·97-s + 0.788·103-s − 0.383·109-s − 0.379·111-s + 0.739·117-s + 6/11·121-s + 0.0887·127-s − 0.704·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354124020\)
\(L(\frac12)\) \(\approx\) \(1.354124020\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ai_bm
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.aq_hy
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ai_gc
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.89.a_adq
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.97.abg_re
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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

−8.707760690069215767720859748121, −8.204992376871640484852414545705, −8.002494181131914373468134125365, −7.34308255208970001455830028366, −6.74739508712700621806548351801, −6.23253373626887200936267143311, −6.01293015988810979454748791954, −5.61895539956247493692671988248, −4.87445902304938786609142494389, −4.23984085275501578963487120772, −3.73709291953889828688513907847, −3.51225399043481939391174543154, −2.20705716352194040660361507275, −1.82857360253644785251609585705, −0.69481716169274455901640598646, 0.69481716169274455901640598646, 1.82857360253644785251609585705, 2.20705716352194040660361507275, 3.51225399043481939391174543154, 3.73709291953889828688513907847, 4.23984085275501578963487120772, 4.87445902304938786609142494389, 5.61895539956247493692671988248, 6.01293015988810979454748791954, 6.23253373626887200936267143311, 6.74739508712700621806548351801, 7.34308255208970001455830028366, 8.002494181131914373468134125365, 8.204992376871640484852414545705, 8.707760690069215767720859748121

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