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Properties

Label 4-777e2-1.1-c1e2-0-37
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  + 3·4-s + 9-s + 5·16-s + 2·25-s + 3·36-s + 6·37-s + 8·47-s + 49-s + 4·53-s + 3·64-s − 8·67-s − 8·71-s − 4·73-s + 81-s + 16·83-s + 6·100-s + 16·101-s + 16·107-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 18·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 3/2·4-s + 1/3·9-s + 5/4·16-s + 2/5·25-s + 1/2·36-s + 0.986·37-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 3/8·64-s − 0.977·67-s − 0.949·71-s − 0.468·73-s + 1/9·81-s + 1.75·83-s + 3/5·100-s + 1.59·101-s + 1.54·107-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 1.47·148-s + 0.0819·149-s + 0.0813·151-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

\(L(1)\) \(\approx\) \(3.156915455\)
\(L(\frac12)\) \(\approx\) \(3.156915455\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
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$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.31.a_abe
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ai_dq
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ae_bu
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.i_fe
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.i_es
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.e_fe
79$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.79.a_bi
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.aq_ig
89$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.89.a_aec
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−8.320021354337537012241642407642, −7.75285386601948514306409613784, −7.44546018727813572031429531776, −7.13635428213251388706733058967, −6.59213111866687116495601119157, −6.17078340898306731077765956330, −5.85964539204743055416890266953, −5.23732941525023925099904976511, −4.62758697612695082516924700800, −4.10706042307528575089429758102, −3.43587267566065340597342015205, −2.86479608663478872177749006124, −2.35227055022536478254567248240, −1.75669575923454840266344870014, −0.924672892020450448822492190869, 0.924672892020450448822492190869, 1.75669575923454840266344870014, 2.35227055022536478254567248240, 2.86479608663478872177749006124, 3.43587267566065340597342015205, 4.10706042307528575089429758102, 4.62758697612695082516924700800, 5.23732941525023925099904976511, 5.85964539204743055416890266953, 6.17078340898306731077765956330, 6.59213111866687116495601119157, 7.13635428213251388706733058967, 7.44546018727813572031429531776, 7.75285386601948514306409613784, 8.320021354337537012241642407642

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