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Properties

Label 4-266240-1.1-c1e2-0-6
Degree $4$
Conductor $266240$
Sign $1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
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  + 3·5-s − 4·9-s + 5·13-s − 6·17-s + 8·25-s + 2·29-s + 4·41-s − 12·45-s + 2·49-s + 4·53-s + 10·61-s + 15·65-s + 8·73-s + 7·81-s − 18·85-s − 14·89-s + 28·97-s + 16·101-s + 20·109-s − 4·113-s − 20·117-s − 8·121-s + 16·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.34·5-s − 4/3·9-s + 1.38·13-s − 1.45·17-s + 8/5·25-s + 0.371·29-s + 0.624·41-s − 1.78·45-s + 2/7·49-s + 0.549·53-s + 1.28·61-s + 1.86·65-s + 0.936·73-s + 7/9·81-s − 1.95·85-s − 1.48·89-s + 2.84·97-s + 1.59·101-s + 1.91·109-s − 0.376·113-s − 1.84·117-s − 0.727·121-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.090845632\)
\(L(\frac12)\) \(\approx\) \(2.090845632\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.19.a_am
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ac_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.43.a_acq
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.47.a_adi
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.59.a_u
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.ak_co
67$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.67.a_c
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ai_ck
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.79.a_aeg
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.o_gw
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.97.abc_ow
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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

−8.851398057444378175871003058714, −8.534523340017778180032981659872, −8.295369517083678724991520514144, −7.41438044784626729527145072063, −6.88492718655749597832350728753, −6.30774625300496195347986505078, −6.09751196615963091653995107475, −5.65544439647962930269474952060, −5.08370929058858175988339217520, −4.54126233583580814862327792516, −3.77224231103807887631189788739, −3.14211894002676362711795503741, −2.43935155545932635516447281163, −1.99162628759937195127750086726, −0.882073877703359510565681406076, 0.882073877703359510565681406076, 1.99162628759937195127750086726, 2.43935155545932635516447281163, 3.14211894002676362711795503741, 3.77224231103807887631189788739, 4.54126233583580814862327792516, 5.08370929058858175988339217520, 5.65544439647962930269474952060, 6.09751196615963091653995107475, 6.30774625300496195347986505078, 6.88492718655749597832350728753, 7.41438044784626729527145072063, 8.295369517083678724991520514144, 8.534523340017778180032981659872, 8.851398057444378175871003058714

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