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Properties

Label 4-756e2-1.1-c1e2-0-15
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  + 2·2-s + 2·4-s + 8·11-s − 4·13-s − 4·16-s + 16·22-s + 12·23-s − 9·25-s − 8·26-s − 8·32-s − 6·37-s + 16·44-s + 24·46-s − 18·47-s + 49-s − 18·50-s − 8·52-s + 30·59-s + 8·61-s − 8·64-s + 24·71-s + 12·73-s − 12·74-s + 18·83-s + 24·92-s − 36·94-s + 24·97-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.41·11-s − 1.10·13-s − 16-s + 3.41·22-s + 2.50·23-s − 9/5·25-s − 1.56·26-s − 1.41·32-s − 0.986·37-s + 2.41·44-s + 3.53·46-s − 2.62·47-s + 1/7·49-s − 2.54·50-s − 1.10·52-s + 3.90·59-s + 1.02·61-s − 64-s + 2.84·71-s + 1.40·73-s − 1.39·74-s + 1.97·83-s + 2.50·92-s − 3.71·94-s + 2.43·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.442583775\)
\(L(\frac12)\) \(\approx\) \(4.442583775\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.41.a_dd
43$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.a_abj
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.47.s_gt
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.59.abe_nf
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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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

−8.650153087392553143775802165667, −7.88452133072696678604809936942, −7.25251494070528755481669441144, −6.75587037635177377501803178294, −6.65813577807847277238815424142, −6.25471251304940803802164013919, −5.40801773340443895405634653905, −5.05971017349345746635432394949, −4.86590526026148497681960970818, −3.90517995805379232478125425669, −3.76204991759879263596774310983, −3.37423655942682476559491319761, −2.42236155159716987191201970387, −1.94425866003935808730361221120, −0.895372083176175841154288549329, 0.895372083176175841154288549329, 1.94425866003935808730361221120, 2.42236155159716987191201970387, 3.37423655942682476559491319761, 3.76204991759879263596774310983, 3.90517995805379232478125425669, 4.86590526026148497681960970818, 5.05971017349345746635432394949, 5.40801773340443895405634653905, 6.25471251304940803802164013919, 6.65813577807847277238815424142, 6.75587037635177377501803178294, 7.25251494070528755481669441144, 7.88452133072696678604809936942, 8.650153087392553143775802165667

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