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Properties

Label 4-777e2-1.1-c1e2-0-17
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  + 9-s − 4·16-s − 7·25-s + 6·37-s + 12·41-s − 10·47-s + 49-s + 4·53-s + 10·67-s + 4·71-s + 14·73-s + 81-s − 14·83-s + 10·101-s + 4·107-s − 21·121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 19·169-s + ⋯
L(s)  = 1  + 1/3·9-s − 16-s − 7/5·25-s + 0.986·37-s + 1.87·41-s − 1.45·47-s + 1/7·49-s + 0.549·53-s + 1.22·67-s + 0.474·71-s + 1.63·73-s + 1/9·81-s − 1.53·83-s + 0.995·101-s + 0.386·107-s − 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.46·169-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

\(L(1)\) \(\approx\) \(1.654379299\)
\(L(\frac12)\) \(\approx\) \(1.654379299\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
37$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.13.a_at
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2^2$ \( 1 - 51 T^{2} + p^{2} T^{4} \) 2.31.a_abz
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.am_de
43$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.43.a_ak
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.k_dq
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.ae_cj
59$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.59.a_adh
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.61.a_de
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.67.ak_fz
71$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.71.ae_fh
73$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.ao_go
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.o_gk
89$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.89.a_acj
97$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.97.a_bp
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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

−8.275962740742164023312994185986, −7.924602239188114007034791987426, −7.62188840177009772059906461732, −7.01172710086903825886282970247, −6.60372622295106809811990187394, −6.19221657725493094835507177119, −5.65569314879668610364825861062, −5.17544341556452146992966082197, −4.57233251622871343091995902414, −4.10849925567150194832561143305, −3.73476206945871048186921604748, −2.88862751607288962954881137888, −2.32269690573043982473290931269, −1.72966379433725597911788631955, −0.65308571840755367172190793075, 0.65308571840755367172190793075, 1.72966379433725597911788631955, 2.32269690573043982473290931269, 2.88862751607288962954881137888, 3.73476206945871048186921604748, 4.10849925567150194832561143305, 4.57233251622871343091995902414, 5.17544341556452146992966082197, 5.65569314879668610364825861062, 6.19221657725493094835507177119, 6.60372622295106809811990187394, 7.01172710086903825886282970247, 7.62188840177009772059906461732, 7.924602239188114007034791987426, 8.275962740742164023312994185986

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