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Properties

Label 4-864e2-1.1-c1e2-0-7
Degree $4$
Conductor $746496$
Sign $1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
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  + 9·11-s − 3·17-s + 8·19-s − 7·25-s − 3·41-s − 43-s + 8·49-s − 9·59-s + 2·67-s − 8·73-s + 3·83-s + 15·89-s + 4·97-s + 15·107-s + 12·113-s + 41·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯
L(s)  = 1  + 2.71·11-s − 0.727·17-s + 1.83·19-s − 7/5·25-s − 0.468·41-s − 0.152·43-s + 8/7·49-s − 1.17·59-s + 0.244·67-s − 0.936·73-s + 0.329·83-s + 1.58·89-s + 0.406·97-s + 1.45·107-s + 1.12·113-s + 3.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.489333142\)
\(L(\frac12)\) \(\approx\) \(2.489333142\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.11.aj_bo
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.23.a_abd
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.d_de
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.b_co
47$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.47.a_ar
53$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.53.a_dh
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.j_eo
61$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.61.a_cy
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ac_ew
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.i_s
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.79.a_abg
83$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.ad_ei
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ap_gw
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ae_gg
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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

−8.408662091648752926941330344616, −7.69121581818258365467822488206, −7.38037533052810416322064368118, −6.97413843062814721922727444024, −6.45397411679532678997968669401, −6.08995721924778924494450951665, −5.71318504436575063198891984101, −5.01540613056604093500846401409, −4.47193239676393967039811169316, −4.01730855827830996627665748837, −3.53774078437030155205676381498, −3.13751370071269219027009949474, −2.10569961098130503324307738358, −1.56891115950101455483309426045, −0.837371603364737429119284908994, 0.837371603364737429119284908994, 1.56891115950101455483309426045, 2.10569961098130503324307738358, 3.13751370071269219027009949474, 3.53774078437030155205676381498, 4.01730855827830996627665748837, 4.47193239676393967039811169316, 5.01540613056604093500846401409, 5.71318504436575063198891984101, 6.08995721924778924494450951665, 6.45397411679532678997968669401, 6.97413843062814721922727444024, 7.38037533052810416322064368118, 7.69121581818258365467822488206, 8.408662091648752926941330344616

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