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Properties

Label 4-2442e2-1.1-c1e2-0-8
Degree $4$
Conductor $5963364$
Sign $-1$
Analytic cond. $380.229$
Root an. cond. $4.41582$
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  + 2·3-s − 4-s − 4·5-s + 3·9-s − 4·11-s − 2·12-s − 8·15-s + 16-s + 4·20-s − 4·23-s + 2·25-s + 4·27-s − 4·31-s − 8·33-s − 3·36-s + 2·37-s + 4·44-s − 12·45-s − 8·47-s + 2·48-s + 2·49-s − 4·53-s + 16·55-s − 4·59-s + 8·60-s − 64-s − 16·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 1.78·5-s + 9-s − 1.20·11-s − 0.577·12-s − 2.06·15-s + 1/4·16-s + 0.894·20-s − 0.834·23-s + 2/5·25-s + 0.769·27-s − 0.718·31-s − 1.39·33-s − 1/2·36-s + 0.328·37-s + 0.603·44-s − 1.78·45-s − 1.16·47-s + 0.288·48-s + 2/7·49-s − 0.549·53-s + 2.15·55-s − 0.520·59-s + 1.03·60-s − 1/8·64-s − 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5963364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5963364\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(380.229\)
Root analytic conductor: \(4.41582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.e_ck
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
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$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.e_abi
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.61.a_de
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.ay_kk
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.73.a_be
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.83.a_bm
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.am_cs
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.abc_ok
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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

−7.40325812522328116162668582788, −6.80031935139358593306537775320, −6.25277707540942830552835496228, −5.86743793231189124007806721872, −5.17573499202240282151979119730, −4.90816884594787275867110960012, −4.37070103961922857699940668084, −4.08334383981651600882495882066, −3.57164188739863552558476287768, −3.35428295593229392000224975162, −2.86751256274482305221434995577, −2.15925311839585975584634470979, −1.77795682771183839383983003699, −0.67380715926683428669346712561, 0, 0.67380715926683428669346712561, 1.77795682771183839383983003699, 2.15925311839585975584634470979, 2.86751256274482305221434995577, 3.35428295593229392000224975162, 3.57164188739863552558476287768, 4.08334383981651600882495882066, 4.37070103961922857699940668084, 4.90816884594787275867110960012, 5.17573499202240282151979119730, 5.86743793231189124007806721872, 6.25277707540942830552835496228, 6.80031935139358593306537775320, 7.40325812522328116162668582788

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