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Properties

Label 4-12e6-1.1-c1e2-0-14
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
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-s + 9-s + 6·11-s + 7·17-s − 3·19-s + 2·25-s − 27-s − 6·33-s + 5·41-s + 6·43-s + 12·49-s − 7·51-s + 3·57-s + 6·59-s − 3·67-s − 2·73-s − 2·75-s + 81-s + 3·83-s − 3·89-s + 20·97-s + 6·99-s + 24·107-s − 17·113-s + 9·121-s − 5·123-s + 127-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.80·11-s + 1.69·17-s − 0.688·19-s + 2/5·25-s − 0.192·27-s − 1.04·33-s + 0.780·41-s + 0.914·43-s + 12/7·49-s − 0.980·51-s + 0.397·57-s + 0.781·59-s − 0.366·67-s − 0.234·73-s − 0.230·75-s + 1/9·81-s + 0.329·83-s − 0.317·89-s + 2.03·97-s + 0.603·99-s + 2.32·107-s − 1.59·113-s + 9/11·121-s − 0.450·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.604160138\)
\(L(\frac12)\) \(\approx\) \(2.604160138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.7.a_am
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.11.ag_bb
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.13.a_p
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ah_bs
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.d_k
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.af_cq
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ag_di
47$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.47.a_x
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.53.a_ace
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.59.ag_dy
61$C_2^2$ \( 1 - 87 T^{2} + p^{2} T^{4} \) 2.61.a_adj
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.d_eu
71$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.71.a_bv
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.c_co
79$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.79.a_bg
83$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.ad_fi
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.d_cs
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.97.au_kz
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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

−7.31223106058867353525896226860, −7.24254311670237849119573991962, −6.71654155580355119025432969652, −6.20507519555421431399393840225, −5.92407596023960430986676223883, −5.62911100482890732889840292480, −5.01333722030571743734298653293, −4.53822653777690835873285023150, −4.06484395206324683883123263241, −3.73159570119362893009333688375, −3.24700755921982229163939881433, −2.53182481403187310711377765049, −1.89086257830681134356215499139, −1.14889945150025819819715670316, −0.76867756404403397827169302035, 0.76867756404403397827169302035, 1.14889945150025819819715670316, 1.89086257830681134356215499139, 2.53182481403187310711377765049, 3.24700755921982229163939881433, 3.73159570119362893009333688375, 4.06484395206324683883123263241, 4.53822653777690835873285023150, 5.01333722030571743734298653293, 5.62911100482890732889840292480, 5.92407596023960430986676223883, 6.20507519555421431399393840225, 6.71654155580355119025432969652, 7.24254311670237849119573991962, 7.31223106058867353525896226860

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