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Properties

Label 4-2944-1.1-c1e2-0-2
Degree $4$
Conductor $2944$
Sign $1$
Analytic cond. $0.187711$
Root an. cond. $0.658222$
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  + 2-s + 4-s − 6·7-s + 8-s − 9-s − 6·14-s + 16-s + 3·17-s − 18-s + 3·23-s − 7·25-s − 6·28-s + 9·31-s + 32-s + 3·34-s − 36-s + 7·41-s + 3·46-s − 3·47-s + 13·49-s − 7·50-s − 6·56-s + 9·62-s + 6·63-s + 64-s + 3·68-s − 10·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 2.26·7-s + 0.353·8-s − 1/3·9-s − 1.60·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.625·23-s − 7/5·25-s − 1.13·28-s + 1.61·31-s + 0.176·32-s + 0.514·34-s − 1/6·36-s + 1.09·41-s + 0.442·46-s − 0.437·47-s + 13/7·49-s − 0.989·50-s − 0.801·56-s + 1.14·62-s + 0.755·63-s + 1/8·64-s + 0.363·68-s − 1.18·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2944\)    =    \(2^{7} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.187711\)
Root analytic conductor: \(0.658222\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8936362456\)
\(L(\frac12)\) \(\approx\) \(0.8936362456\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.11.a_am
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ad_g
19$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.19.a_abg
29$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.29.a_q
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.aj_ca
37$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.37.a_aby
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ah_cm
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.d_co
53$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.53.a_bn
59$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.59.a_ar
61$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.61.a_bd
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.67.a_abu
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.k_bu
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.g_bi
79$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.f_fe
83$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.83.a_ae
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.h_ge
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.c_cw
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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

−12.82689379624976692935685457416, −12.47082892348308743685163307187, −11.78360452102593778329276783241, −11.24783149837249233207705572275, −10.21693082504046408147657388629, −9.926115118339504026757819192086, −9.356657181464702211244177641224, −8.481057393643706361731588675868, −7.56256187415924558038549191686, −6.85258559439377508167524060896, −6.10715892759699976006896801110, −5.77186601537372124347663154498, −4.47999875102112923550436568355, −3.44337038071032074498714277783, −2.82824021742699524660793330370, 2.82824021742699524660793330370, 3.44337038071032074498714277783, 4.47999875102112923550436568355, 5.77186601537372124347663154498, 6.10715892759699976006896801110, 6.85258559439377508167524060896, 7.56256187415924558038549191686, 8.481057393643706361731588675868, 9.356657181464702211244177641224, 9.926115118339504026757819192086, 10.21693082504046408147657388629, 11.24783149837249233207705572275, 11.78360452102593778329276783241, 12.47082892348308743685163307187, 12.82689379624976692935685457416

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