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Properties

Label 4-5227200-1.1-c1e2-0-29
Degree $4$
Conductor $5227200$
Sign $1$
Analytic cond. $333.290$
Root an. cond. $4.27273$
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 + 4·5-s + 9-s − 2·11-s + 4·15-s − 6·23-s + 11·25-s + 27-s − 8·31-s − 2·33-s + 2·37-s + 4·45-s − 2·47-s + 8·49-s + 14·53-s − 8·55-s + 8·59-s + 14·67-s − 6·69-s + 11·75-s + 81-s − 8·89-s − 8·93-s − 18·97-s − 2·99-s + 10·103-s + 2·111-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 1.03·15-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 1.43·31-s − 0.348·33-s + 0.328·37-s + 0.596·45-s − 0.291·47-s + 8/7·49-s + 1.92·53-s − 1.07·55-s + 1.04·59-s + 1.71·67-s − 0.722·69-s + 1.27·75-s + 1/9·81-s − 0.847·89-s − 0.829·93-s − 1.82·97-s − 0.201·99-s + 0.985·103-s + 0.189·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5227200\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(333.290\)
Root analytic conductor: \(4.27273\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5227200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.290504606\)
\(L(\frac12)\) \(\approx\) \(4.290504606\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.17.a_i
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.g_bu
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.i_bq
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ac_by
41$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.41.a_abm
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.43.a_q
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.c_bu
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ao_fa
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.61.a_abi
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.ao_gs
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.73.a_acm
79$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.79.a_acc
83$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.83.a_ay
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.i_gw
97$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.s_ko
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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

−7.22081445821323812185703917019, −6.84339293447776337810667502250, −6.59068565317438925155567612097, −5.90011308306553386127242469516, −5.66299449995493948247146288710, −5.42690899293069439365063838654, −4.97090328020147701564370881997, −4.32125063493921347349656123492, −3.90946734191003820640835947158, −3.44258566124391067400014378658, −2.73044273296147154407240924696, −2.37859158699726680228149915057, −2.03340786319214331073935928918, −1.50449458536836257081068089117, −0.66894689054323161210674966401, 0.66894689054323161210674966401, 1.50449458536836257081068089117, 2.03340786319214331073935928918, 2.37859158699726680228149915057, 2.73044273296147154407240924696, 3.44258566124391067400014378658, 3.90946734191003820640835947158, 4.32125063493921347349656123492, 4.97090328020147701564370881997, 5.42690899293069439365063838654, 5.66299449995493948247146288710, 5.90011308306553386127242469516, 6.59068565317438925155567612097, 6.84339293447776337810667502250, 7.22081445821323812185703917019

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