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Properties

Label 4-338688-1.1-c1e2-0-20
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 − 6·25-s + 27-s + 12·37-s + 16·47-s − 7·49-s + 8·59-s − 6·75-s + 81-s + 24·83-s − 20·109-s + 12·111-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 16·141-s − 7·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 6/5·25-s + 0.192·27-s + 1.97·37-s + 2.33·47-s − 49-s + 1.04·59-s − 0.692·75-s + 1/9·81-s + 2.63·83-s − 1.91·109-s + 1.13·111-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.34·141-s − 0.577·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-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

\(L(1)\) \(\approx\) \(2.208428044\)
\(L(\frac12)\) \(\approx\) \(2.208428044\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.71.a_abq
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \) 2.97.a_agw
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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

−8.819232585646660215792784507209, −8.193528111584429528608742769278, −7.81088115160325207251355213223, −7.57707322471367984261875892913, −6.91773934502363544492277410633, −6.42220622746433728521228673103, −5.94035980383987971544431680774, −5.43719813555882240430234597605, −4.84061392476483004743428124518, −4.09999936954950450597003806766, −3.93195351568068559676029275292, −3.09477818906188854039088131353, −2.48817396146028453358802572710, −1.90950318412374428321071078017, −0.839819930261735239821289741006, 0.839819930261735239821289741006, 1.90950318412374428321071078017, 2.48817396146028453358802572710, 3.09477818906188854039088131353, 3.93195351568068559676029275292, 4.09999936954950450597003806766, 4.84061392476483004743428124518, 5.43719813555882240430234597605, 5.94035980383987971544431680774, 6.42220622746433728521228673103, 6.91773934502363544492277410633, 7.57707322471367984261875892913, 7.81088115160325207251355213223, 8.193528111584429528608742769278, 8.819232585646660215792784507209

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