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Properties

Label 4-560e2-1.1-c1e2-0-10
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 7-s + 3·9-s − 6·11-s − 8·13-s + 15-s + 2·19-s + 21-s − 3·23-s + 8·27-s − 6·29-s + 8·31-s − 6·33-s + 35-s + 4·37-s − 8·39-s + 18·41-s + 14·43-s + 3·45-s − 6·49-s + 6·53-s − 6·55-s + 2·57-s − 6·59-s − 5·61-s + 3·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 9-s − 1.80·11-s − 2.21·13-s + 0.258·15-s + 0.458·19-s + 0.218·21-s − 0.625·23-s + 1.53·27-s − 1.11·29-s + 1.43·31-s − 1.04·33-s + 0.169·35-s + 0.657·37-s − 1.28·39-s + 2.81·41-s + 2.13·43-s + 0.447·45-s − 6/7·49-s + 0.824·53-s − 0.809·55-s + 0.264·57-s − 0.781·59-s − 0.640·61-s + 0.377·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141648311\)
\(L(\frac12)\) \(\approx\) \(2.141648311\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_z
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_av
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.43.ao_ff
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ax
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_hb
79$C_2^2$ \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_acx
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.83.g_gt
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−10.82465588357168608819917333454, −10.52610291090482988796514097006, −9.950990715049745388256337962014, −9.737916287566358540414165534832, −9.345121989817046898139113135481, −8.954531385528270619857509295449, −7.958895390944305290544764811325, −7.84031517987816893486907137675, −7.66773938004256374424419296030, −7.13977471800601991289190397013, −6.49011580126593528066955492493, −5.88088910561294881195022216836, −5.30905693630335057621821325933, −4.88254824613117505382671804894, −4.49408495836826359745648464830, −3.88345065625122588213397001023, −2.80063679728376044789761690869, −2.51464033325386093398727874030, −2.13931140405510732205263691279, −0.811805894555582142530315413422, 0.811805894555582142530315413422, 2.13931140405510732205263691279, 2.51464033325386093398727874030, 2.80063679728376044789761690869, 3.88345065625122588213397001023, 4.49408495836826359745648464830, 4.88254824613117505382671804894, 5.30905693630335057621821325933, 5.88088910561294881195022216836, 6.49011580126593528066955492493, 7.13977471800601991289190397013, 7.66773938004256374424419296030, 7.84031517987816893486907137675, 7.958895390944305290544764811325, 8.954531385528270619857509295449, 9.345121989817046898139113135481, 9.737916287566358540414165534832, 9.950990715049745388256337962014, 10.52610291090482988796514097006, 10.82465588357168608819917333454

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