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Properties

Label 4-120704-1.1-c1e2-0-3
Degree $4$
Conductor $120704$
Sign $1$
Analytic cond. $7.69619$
Root an. cond. $1.66559$
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 + 7·7-s + 8-s − 3·9-s + 7·14-s + 16-s − 17-s − 3·18-s + 7·23-s − 4·25-s + 7·28-s − 2·31-s + 32-s − 34-s − 3·36-s − 4·41-s + 7·46-s + 3·47-s + 23·49-s − 4·50-s + 7·56-s − 2·62-s − 21·63-s + 64-s − 68-s + 17·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 2.64·7-s + 0.353·8-s − 9-s + 1.87·14-s + 1/4·16-s − 0.242·17-s − 0.707·18-s + 1.45·23-s − 4/5·25-s + 1.32·28-s − 0.359·31-s + 0.176·32-s − 0.171·34-s − 1/2·36-s − 0.624·41-s + 1.03·46-s + 0.437·47-s + 23/7·49-s − 0.565·50-s + 0.935·56-s − 0.254·62-s − 2.64·63-s + 1/8·64-s − 0.121·68-s + 2.01·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(120704\)    =    \(2^{7} \cdot 23 \cdot 41\)
Sign: $1$
Analytic conductor: \(7.69619\)
Root analytic conductor: \(1.66559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 120704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.121586919\)
\(L(\frac12)\) \(\approx\) \(3.121586919\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.7.ah_ba
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.b_bc
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.c_ch
37$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.37.a_bs
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.43.a_aq
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.ad_bo
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.59.a_ai
61$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.61.a_bw
67$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.67.a_cj
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.71.ar_ie
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.m_gw
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ae_gc
83$C_2^2$ \( 1 + 142 T^{2} + p^{2} T^{4} \) 2.83.a_fm
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.c_abu
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.p_gw
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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

−9.331449026030771788897144311019, −8.744577263423261702457651837406, −8.371275623099769647878206754613, −8.047430172666790448873924436398, −7.45919763612733122921362489529, −7.07616405946511270265034631607, −6.30016863780431003916917134097, −5.66365149740483766686267954769, −5.17675246504283970063759802130, −4.92064750501566655418749676394, −4.31860383057599979992277838011, −3.63109002150976200640855375019, −2.74801659505619589886435193376, −2.05880965270091163799051309522, −1.31454357392874755931120654964, 1.31454357392874755931120654964, 2.05880965270091163799051309522, 2.74801659505619589886435193376, 3.63109002150976200640855375019, 4.31860383057599979992277838011, 4.92064750501566655418749676394, 5.17675246504283970063759802130, 5.66365149740483766686267954769, 6.30016863780431003916917134097, 7.07616405946511270265034631607, 7.45919763612733122921362489529, 8.047430172666790448873924436398, 8.371275623099769647878206754613, 8.744577263423261702457651837406, 9.331449026030771788897144311019

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