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Properties

Label 4-2376e2-1.1-c1e2-0-4
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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 − 5-s + 9-s + 11-s + 15-s + 11·23-s − 7·25-s − 27-s + 31-s − 33-s + 6·37-s − 45-s + 4·49-s − 7·53-s − 55-s + 5·59-s + 7·67-s − 11·69-s − 14·71-s + 7·75-s + 81-s + 3·89-s − 93-s − 2·97-s + 99-s − 13·103-s − 6·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.258·15-s + 2.29·23-s − 7/5·25-s − 0.192·27-s + 0.179·31-s − 0.174·33-s + 0.986·37-s − 0.149·45-s + 4/7·49-s − 0.961·53-s − 0.134·55-s + 0.650·59-s + 0.855·67-s − 1.32·69-s − 1.66·71-s + 0.808·75-s + 1/9·81-s + 0.317·89-s − 0.103·93-s − 0.203·97-s + 0.100·99-s − 1.28·103-s − 0.569·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568721148\)
\(L(\frac12)\) \(\approx\) \(1.568721148\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
11$C_2$ \( 1 - T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.b_i
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.17.a_y
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.al_cw
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ab_bq
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.37.ag_cw
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.41.a_ai
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.43.a_q
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.h_em
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.59.af_es
61$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.61.a_bf
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.ah_fo
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.o_ha
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.79.a_k
83$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.83.a_aek
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.ad_ge
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.c_abe
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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

−7.28119764534007400216802222500, −6.74002633709198782859618207613, −6.58301968070034290546810553082, −6.06522983806348728701560321268, −5.54614024262158236277620538250, −5.33405676904630758582409113094, −4.78173985431688540235657488566, −4.32568663019540457795320183471, −4.06198566532748267900073665011, −3.44258669583546492318573690658, −3.00445962601174157046498950512, −2.49280164357522281591384073530, −1.76215085138594824348608165197, −1.15430087372348501427781491541, −0.50003608058446863513391263976, 0.50003608058446863513391263976, 1.15430087372348501427781491541, 1.76215085138594824348608165197, 2.49280164357522281591384073530, 3.00445962601174157046498950512, 3.44258669583546492318573690658, 4.06198566532748267900073665011, 4.32568663019540457795320183471, 4.78173985431688540235657488566, 5.33405676904630758582409113094, 5.54614024262158236277620538250, 6.06522983806348728701560321268, 6.58301968070034290546810553082, 6.74002633709198782859618207613, 7.28119764534007400216802222500

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