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Properties

Label 4-4240-1.1-c1e2-0-0
Degree $4$
Conductor $4240$
Sign $1$
Analytic cond. $0.270346$
Root an. cond. $0.721074$
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  − 2·4-s + 5-s + 4·9-s + 4·13-s + 4·16-s − 12·17-s − 2·20-s − 4·25-s − 8·36-s − 2·37-s − 6·41-s + 4·45-s + 2·49-s − 8·52-s − 13·53-s + 10·61-s − 8·64-s + 4·65-s + 24·68-s − 2·73-s + 4·80-s + 7·81-s − 12·85-s + 24·89-s + 10·97-s + 8·100-s − 12·101-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 4/3·9-s + 1.10·13-s + 16-s − 2.91·17-s − 0.447·20-s − 4/5·25-s − 4/3·36-s − 0.328·37-s − 0.937·41-s + 0.596·45-s + 2/7·49-s − 1.10·52-s − 1.78·53-s + 1.28·61-s − 64-s + 0.496·65-s + 2.91·68-s − 0.234·73-s + 0.447·80-s + 7/9·81-s − 1.30·85-s + 2.54·89-s + 1.01·97-s + 4/5·100-s − 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4240\)    =    \(2^{4} \cdot 5 \cdot 53\)
Sign: $1$
Analytic conductor: \(0.270346\)
Root analytic conductor: \(0.721074\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4240,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7441133267\)
\(L(\frac12)\) \(\approx\) \(0.7441133267\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 12 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.23.a_q
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.c_co
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.47.a_bi
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.ak_fi
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.67.a_dw
71$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.71.a_adi
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.c_co
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.83.a_e
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.ay_la
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ak_ic
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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

−12.73091995435072935616474322247, −11.87695953386478830377450176328, −11.12861149270607666911621761179, −10.63530562675408550669833007924, −9.979551782259052851675936756200, −9.414247610556808307059138589484, −8.832715740821302498668790162905, −8.389143546910867311269051532984, −7.46243451280049599709785862653, −6.62566816337418312191905909544, −6.14324073284337730711567884401, −4.99155582763338176726655577441, −4.38568537084502361576790861730, −3.68324386515125233033208675776, −1.90044597069967539028129303549, 1.90044597069967539028129303549, 3.68324386515125233033208675776, 4.38568537084502361576790861730, 4.99155582763338176726655577441, 6.14324073284337730711567884401, 6.62566816337418312191905909544, 7.46243451280049599709785862653, 8.389143546910867311269051532984, 8.832715740821302498668790162905, 9.414247610556808307059138589484, 9.979551782259052851675936756200, 10.63530562675408550669833007924, 11.12861149270607666911621761179, 11.87695953386478830377450176328, 12.73091995435072935616474322247

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