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Properties

Label 4-266240-1.1-c1e2-0-7
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  + 5-s + 4·9-s + 3·13-s − 6·17-s − 4·25-s + 6·29-s − 4·37-s + 4·45-s + 14·49-s + 2·61-s + 3·65-s + 4·73-s + 7·81-s − 6·85-s + 6·89-s + 16·97-s + 24·101-s − 4·109-s + 24·113-s + 12·117-s − 4·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 6·145-s + ⋯
L(s)  = 1  + 0.447·5-s + 4/3·9-s + 0.832·13-s − 1.45·17-s − 4/5·25-s + 1.11·29-s − 0.657·37-s + 0.596·45-s + 2·49-s + 0.256·61-s + 0.372·65-s + 0.468·73-s + 7/9·81-s − 0.650·85-s + 0.635·89-s + 1.62·97-s + 2.38·101-s − 0.383·109-s + 2.25·113-s + 1.10·117-s − 0.363·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.498·145-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

\(L(1)\) \(\approx\) \(2.161634376\)
\(L(\frac12)\) \(\approx\) \(2.161634376\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + p T^{2} ) \)
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_ae
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.43.a_cy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.ac_bq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ag_ec
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.852548028896646939818017728608, −8.634026819554766217841717310753, −7.939793307915262172985723458981, −7.42129257029188555217003472189, −7.01161354177442620618975331971, −6.49738799354723941084853818562, −6.15053656536159823411456492835, −5.57964335350261823398441871429, −4.84243133357058012361273072667, −4.45466497421094809618038653383, −3.91334227850857666254892750975, −3.35333552543317344686467621778, −2.33153255264834640750877024823, −1.89954000762703741088074067599, −0.931417515152989032775979047311, 0.931417515152989032775979047311, 1.89954000762703741088074067599, 2.33153255264834640750877024823, 3.35333552543317344686467621778, 3.91334227850857666254892750975, 4.45466497421094809618038653383, 4.84243133357058012361273072667, 5.57964335350261823398441871429, 6.15053656536159823411456492835, 6.49738799354723941084853818562, 7.01161354177442620618975331971, 7.42129257029188555217003472189, 7.939793307915262172985723458981, 8.634026819554766217841717310753, 8.852548028896646939818017728608

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