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Properties

Label 4-2442e2-1.1-c1e2-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 2·5-s + 3·9-s + 2·11-s + 2·12-s + 4·15-s + 16-s + 2·20-s + 4·23-s − 3·25-s + 4·27-s + 6·31-s + 4·33-s + 3·36-s + 7·37-s + 2·44-s + 6·45-s + 10·47-s + 2·48-s + 2·49-s + 53-s + 4·55-s − 13·59-s + 4·60-s + 64-s − 3·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s + 0.603·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s + 0.447·20-s + 0.834·23-s − 3/5·25-s + 0.769·27-s + 1.07·31-s + 0.696·33-s + 1/2·36-s + 1.15·37-s + 0.301·44-s + 0.894·45-s + 1.45·47-s + 0.288·48-s + 2/7·49-s + 0.137·53-s + 0.539·55-s − 1.69·59-s + 0.516·60-s + 1/8·64-s − 0.366·67-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: \(0\)
Selberg data: \((4,\ 5963364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.901501585\)
\(L(\frac12)\) \(\approx\) \(6.901501585\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
37$C_2$ \( 1 - 7 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.ac_h
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.17.a_ad
19$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.19.a_l
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ae_bu
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.31.ag_cp
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.41.a_ai
43$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.43.a_abh
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.ak_eo
53$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.ab_ae
59$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.59.n_dk
61$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.61.a_c
67$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.d_fg
71$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ah_fm
73$C_2^2$ \( 1 + 127 T^{2} + p^{2} T^{4} \) 2.73.a_ex
79$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.79.a_ar
83$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.83.a_t
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.c_k
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ah_em
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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

−7.34684927560729861514725628022, −6.71579345103270080101970476895, −6.56936323830077869054928037796, −6.11761745905761028272819763338, −5.65560689632968390875720589841, −5.32427500319042752171811199330, −4.61961497473907061220675551614, −4.27909820458333652055016570791, −3.89890218861491613308323060927, −3.20601237731975112704371048865, −2.92199779645343700804814576141, −2.40444536306630366973006394895, −2.00323295860235314380955774321, −1.41869905548819710932692919181, −0.854076139755655987602375655313, 0.854076139755655987602375655313, 1.41869905548819710932692919181, 2.00323295860235314380955774321, 2.40444536306630366973006394895, 2.92199779645343700804814576141, 3.20601237731975112704371048865, 3.89890218861491613308323060927, 4.27909820458333652055016570791, 4.61961497473907061220675551614, 5.32427500319042752171811199330, 5.65560689632968390875720589841, 6.11761745905761028272819763338, 6.56936323830077869054928037796, 6.71579345103270080101970476895, 7.34684927560729861514725628022

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