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Properties

Label 4-338688-1.1-c1e2-0-23
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 − 2·5-s + 2·7-s + 9-s − 2·15-s + 10·17-s + 2·21-s − 6·25-s + 27-s − 4·35-s + 4·37-s + 10·41-s − 4·43-s − 2·45-s − 4·47-s − 3·49-s + 10·51-s + 16·59-s + 2·63-s + 20·67-s − 6·75-s − 12·79-s + 81-s − 12·83-s − 20·85-s + 2·89-s + 6·101-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.516·15-s + 2.42·17-s + 0.436·21-s − 6/5·25-s + 0.192·27-s − 0.676·35-s + 0.657·37-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s − 3/7·49-s + 1.40·51-s + 2.08·59-s + 0.251·63-s + 2.44·67-s − 0.692·75-s − 1.35·79-s + 1/9·81-s − 1.31·83-s − 2.16·85-s + 0.211·89-s + 0.597·101-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

\(L(1)\) \(\approx\) \(2.227787068\)
\(L(\frac12)\) \(\approx\) \(2.227787068\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.c_k
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.ak_cg
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.23.a_ag
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ak_du
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.e_ba
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.e_ck
53$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.53.a_ck
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.au_iw
71$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.71.a_abu
73$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.73.a_ao
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.m_hi
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.m_gk
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.ac_k
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−8.473682014229751997405731452080, −8.224038304596750028251763592796, −7.86177920742879026993859900497, −7.57906802187173278880300395572, −7.12244226827991771808405363934, −6.47898459061222374366557338066, −5.71594502551140744667761514086, −5.47997274244345254376344070134, −4.83635502235543271892438016303, −4.13193994245512421721334201867, −3.80657942962184764212042185110, −3.26337563107656564785483163957, −2.56755017168898129584163300871, −1.72766775935589925050008072415, −0.883344151989972142699671395578, 0.883344151989972142699671395578, 1.72766775935589925050008072415, 2.56755017168898129584163300871, 3.26337563107656564785483163957, 3.80657942962184764212042185110, 4.13193994245512421721334201867, 4.83635502235543271892438016303, 5.47997274244345254376344070134, 5.71594502551140744667761514086, 6.47898459061222374366557338066, 7.12244226827991771808405363934, 7.57906802187173278880300395572, 7.86177920742879026993859900497, 8.224038304596750028251763592796, 8.473682014229751997405731452080

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