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Properties

Label 4-2944-1.1-c1e2-0-0
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 + 4·7-s − 8-s − 5·9-s − 4·14-s + 16-s − 3·17-s + 5·18-s + 5·23-s − 25-s + 4·28-s − 5·31-s − 32-s + 3·34-s − 5·36-s − 15·41-s − 5·46-s − 3·47-s + 7·49-s + 50-s − 4·56-s + 5·62-s − 20·63-s + 64-s − 3·68-s + 6·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 1.17·18-s + 1.04·23-s − 1/5·25-s + 0.755·28-s − 0.898·31-s − 0.176·32-s + 0.514·34-s − 5/6·36-s − 2.34·41-s − 0.737·46-s − 0.437·47-s + 49-s + 0.141·50-s − 0.534·56-s + 0.635·62-s − 2.51·63-s + 1/8·64-s − 0.363·68-s + 0.712·71-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

\(L(1)\) \(\approx\) \(0.5594681166\)
\(L(\frac12)\) \(\approx\) \(0.5594681166\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
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$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.f_bw
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.p_fg
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.43.a_acq
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.d_dq
53$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.53.a_ax
59$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.59.a_dn
61$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.61.a_af
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.67.a_bu
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ag_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.ak_dm
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.ah_fu
83$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.83.a_ca
89$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.d_ge
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ak_fi
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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

−12.78912282019567903027660297416, −11.83783159581439458116776577470, −11.61427577959493971618095536628, −10.95456420545025768805091152100, −10.74399486704281593022944103063, −9.682467835415954696911097446948, −8.917722921850910464016775230536, −8.493217092368260651723415382850, −8.048385928525224176489037386795, −7.20315367744419609328011967740, −6.38172203634760313512956515328, −5.40491758132966709594557643401, −4.86576179293298387663308554451, −3.35991069196061239692692583694, −2.06813763014627359933851385827, 2.06813763014627359933851385827, 3.35991069196061239692692583694, 4.86576179293298387663308554451, 5.40491758132966709594557643401, 6.38172203634760313512956515328, 7.20315367744419609328011967740, 8.048385928525224176489037386795, 8.493217092368260651723415382850, 8.917722921850910464016775230536, 9.682467835415954696911097446948, 10.74399486704281593022944103063, 10.95456420545025768805091152100, 11.61427577959493971618095536628, 11.83783159581439458116776577470, 12.78912282019567903027660297416

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