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Properties

Label 4-338688-1.1-c1e2-0-38
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 + 11-s + 7·13-s + 10·23-s − 25-s + 27-s + 33-s − 7·37-s + 7·39-s + 22·47-s + 49-s − 5·59-s + 61-s + 10·69-s − 18·71-s − 18·73-s − 75-s + 81-s − 83-s − 7·97-s + 99-s + 15·107-s − 15·109-s − 7·111-s + 7·117-s − 15·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.94·13-s + 2.08·23-s − 1/5·25-s + 0.192·27-s + 0.174·33-s − 1.15·37-s + 1.12·39-s + 3.20·47-s + 1/7·49-s − 0.650·59-s + 0.128·61-s + 1.20·69-s − 2.13·71-s − 2.10·73-s − 0.115·75-s + 1/9·81-s − 0.109·83-s − 0.710·97-s + 0.100·99-s + 1.45·107-s − 1.43·109-s − 0.664·111-s + 0.647·117-s − 1.36·121-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

\(L(1)\) \(\approx\) \(2.752587289\)
\(L(\frac12)\) \(\approx\) \(2.752587289\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ab_q
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.13.ah_bm
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \) 2.19.a_abb
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.23.ak_ck
29$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.29.a_r
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.h_dg
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.a_abj
47$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.47.aw_id
53$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.a_dd
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.f_ai
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.ab_adk
67$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \) 2.67.a_bb
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.s_gs
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.s_gw
79$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.79.a_dk
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.b_dq
89$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.89.a_aes
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.h_ds
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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

−8.701345258628524799493502891276, −8.579606686909254369339383987202, −7.85115845474126817890955824277, −7.19855738149753283316587676438, −7.12554901525322288267898176274, −6.42957037120311725068238678708, −5.83624567592269771620558714493, −5.58456053521090848271576483985, −4.77365493929464278344570379723, −4.20737628930419033341751170217, −3.75195049695565634703421883124, −3.14563115731238966789018383698, −2.66441852233658563658560463972, −1.61496031156866813415951111125, −1.04300274911208976300053555547, 1.04300274911208976300053555547, 1.61496031156866813415951111125, 2.66441852233658563658560463972, 3.14563115731238966789018383698, 3.75195049695565634703421883124, 4.20737628930419033341751170217, 4.77365493929464278344570379723, 5.58456053521090848271576483985, 5.83624567592269771620558714493, 6.42957037120311725068238678708, 7.12554901525322288267898176274, 7.19855738149753283316587676438, 7.85115845474126817890955824277, 8.579606686909254369339383987202, 8.701345258628524799493502891276

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