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Properties

Label 4-777e2-1.1-c1e2-0-20
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $38.4942$
Root an. cond. $2.49085$
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  − 4-s + 9-s + 4·11-s − 3·16-s − 4·25-s − 36-s − 10·41-s − 4·44-s + 12·47-s − 7·49-s + 4·53-s + 7·64-s − 8·67-s − 8·71-s + 14·73-s + 81-s − 4·83-s + 4·99-s + 4·100-s + 22·101-s + 24·107-s − 6·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/3·9-s + 1.20·11-s − 3/4·16-s − 4/5·25-s − 1/6·36-s − 1.56·41-s − 0.603·44-s + 1.75·47-s − 49-s + 0.549·53-s + 7/8·64-s − 0.977·67-s − 0.949·71-s + 1.63·73-s + 1/9·81-s − 0.439·83-s + 0.402·99-s + 2/5·100-s + 2.18·101-s + 2.32·107-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(603729\)    =    \(3^{2} \cdot 7^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(38.4942\)
Root analytic conductor: \(2.49085\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586453788\)
\(L(\frac12)\) \(\approx\) \(1.586453788\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
37$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.17.a_abc
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.31.a_y
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.k_de
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.43.a_bq
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.am_ew
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.ae_k
59$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.59.a_ci
61$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \) 2.61.a_abw
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.ao_he
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.e_gk
89$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \) 2.89.a_fk
97$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.97.a_acu
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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

−8.622911407201382228901759441203, −7.981674562441776739002986742343, −7.44544023495299927345153264705, −7.06142726980974237559906420435, −6.62553319329423616688177296747, −6.15333457992735215092707896149, −5.69261716591850380686163082689, −5.08555143087979776835219072489, −4.50925468081488809127624716951, −4.23448926333164837222582469995, −3.61643996095770439722418539018, −3.15792286604055879967350136914, −2.17171889635515742619997033871, −1.68831836896583862933344868969, −0.65858084215797769591130251746, 0.65858084215797769591130251746, 1.68831836896583862933344868969, 2.17171889635515742619997033871, 3.15792286604055879967350136914, 3.61643996095770439722418539018, 4.23448926333164837222582469995, 4.50925468081488809127624716951, 5.08555143087979776835219072489, 5.69261716591850380686163082689, 6.15333457992735215092707896149, 6.62553319329423616688177296747, 7.06142726980974237559906420435, 7.44544023495299927345153264705, 7.981674562441776739002986742343, 8.622911407201382228901759441203

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