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Properties

Label 4-272e2-1.1-c1e2-0-37
Degree $4$
Conductor $73984$
Sign $1$
Analytic cond. $4.71728$
Root an. cond. $1.47374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s − 4·7-s + 8·9-s + 4·11-s − 8·13-s + 8·15-s − 2·17-s + 16·21-s − 4·23-s + 2·25-s − 12·27-s − 6·29-s − 12·31-s − 16·33-s + 8·35-s − 10·37-s + 32·39-s + 2·41-s − 16·45-s + 8·49-s + 8·51-s − 8·55-s − 2·61-s − 32·63-s + 16·65-s − 24·67-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s − 1.51·7-s + 8/3·9-s + 1.20·11-s − 2.21·13-s + 2.06·15-s − 0.485·17-s + 3.49·21-s − 0.834·23-s + 2/5·25-s − 2.30·27-s − 1.11·29-s − 2.15·31-s − 2.78·33-s + 1.35·35-s − 1.64·37-s + 5.12·39-s + 0.312·41-s − 2.38·45-s + 8/7·49-s + 1.12·51-s − 1.07·55-s − 0.256·61-s − 4.03·63-s + 1.98·65-s − 2.93·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(73984\)    =    \(2^{8} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(4.71728\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 73984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.c_c
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_i
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_i
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_i
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.g_s
31$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ac_c
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.59.a_acc
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_cu
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_s
79$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_i
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.89.q_ji
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_s
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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

−11.69307476059153574624718103556, −11.39152667188298665814085213858, −10.69938993569114459202053886974, −10.58027560156502004373781830544, −9.794841308445417639763710870410, −9.372042058745258052040311736729, −9.203504505709253444089388755482, −8.167808807301135398038047372684, −7.26958994912748527459196450277, −7.13196773457229752788795391652, −6.78657543972434037225504531405, −6.02340381489693443218112682214, −5.69304938056123144287300980663, −5.16442537799594780126266106217, −4.42870091362551455260846792108, −3.93372419305416020206054009696, −3.22071629164455272397309147559, −1.91285182750641487729136243212, 0, 0, 1.91285182750641487729136243212, 3.22071629164455272397309147559, 3.93372419305416020206054009696, 4.42870091362551455260846792108, 5.16442537799594780126266106217, 5.69304938056123144287300980663, 6.02340381489693443218112682214, 6.78657543972434037225504531405, 7.13196773457229752788795391652, 7.26958994912748527459196450277, 8.167808807301135398038047372684, 9.203504505709253444089388755482, 9.372042058745258052040311736729, 9.794841308445417639763710870410, 10.58027560156502004373781830544, 10.69938993569114459202053886974, 11.39152667188298665814085213858, 11.69307476059153574624718103556

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