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Properties

Label 4-72e3-1.1-c1e2-0-14
Degree $4$
Conductor $373248$
Sign $-1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·25-s + 4·29-s + 8·43-s − 8·47-s − 49-s − 2·53-s + 8·67-s − 24·71-s − 10·73-s + 14·97-s − 18·101-s − 5·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.894·5-s − 3/5·25-s + 0.742·29-s + 1.21·43-s − 1.16·47-s − 1/7·49-s − 0.274·53-s + 0.977·67-s − 2.84·71-s − 1.17·73-s + 1.42·97-s − 1.79·101-s − 0.454·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.c_h
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.31.a_ah
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ai_di
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.i_dq
53$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.c_dz
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ai_fe
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.k_ed
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 - 99 T^{2} + p^{2} T^{4} \) 2.83.a_adv
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.89.a_aek
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ao_dv
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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

−8.439066662257424695112800861791, −7.956436563059469962811587031596, −7.58602992349881920685979934493, −7.20288572346483026200271277989, −6.61737794822918205966998712942, −6.13482733734652566668429030948, −5.65157018566792001471232564127, −5.05477985426488982718417601378, −4.37843421879350425656424477814, −4.17761849076382515525602931682, −3.40001952647623779468470812883, −2.94473239580093011932541074185, −2.13908420032515843764991325320, −1.22054092829665517274409188645, 0, 1.22054092829665517274409188645, 2.13908420032515843764991325320, 2.94473239580093011932541074185, 3.40001952647623779468470812883, 4.17761849076382515525602931682, 4.37843421879350425656424477814, 5.05477985426488982718417601378, 5.65157018566792001471232564127, 6.13482733734652566668429030948, 6.61737794822918205966998712942, 7.20288572346483026200271277989, 7.58602992349881920685979934493, 7.956436563059469962811587031596, 8.439066662257424695112800861791

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