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Properties

Label 4-259308-1.1-c1e2-0-4
Degree $4$
Conductor $259308$
Sign $1$
Analytic cond. $16.5337$
Root an. cond. $2.01647$
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-s − 4-s + 9-s − 12-s + 6·13-s + 16-s + 12·19-s − 2·25-s + 27-s − 4·31-s − 36-s − 8·37-s + 6·39-s + 4·43-s + 48-s − 6·52-s + 12·57-s + 10·61-s − 64-s + 4·67-s + 2·73-s − 2·75-s − 12·76-s − 8·79-s + 81-s − 4·93-s − 14·97-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s + 1.66·13-s + 1/4·16-s + 2.75·19-s − 2/5·25-s + 0.192·27-s − 0.718·31-s − 1/6·36-s − 1.31·37-s + 0.960·39-s + 0.609·43-s + 0.144·48-s − 0.832·52-s + 1.58·57-s + 1.28·61-s − 1/8·64-s + 0.488·67-s + 0.234·73-s − 0.230·75-s − 1.37·76-s − 0.900·79-s + 1/9·81-s − 0.414·93-s − 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(259308\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16.5337\)
Root analytic conductor: \(2.01647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 259308,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.266209564\)
\(L(\frac12)\) \(\approx\) \(2.266209564\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 - T \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ag_bi
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.17.a_ag
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.e_by
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.i_cc
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ae_di
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.53.a_abq
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ak_es
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.ae_acg
71$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.71.a_dm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.ac_co
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.83.a_bm
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.89.a_s
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.o_gg
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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

−8.814921481318144229906760633094, −8.594949072848139512290444008011, −8.045451793964213293582863010431, −7.50830134670769843729117769646, −7.22891664623198580869388782053, −6.58418622739578159912508579609, −5.95634878468840258125794611097, −5.39281692481958639495063470626, −5.18278653808789891360302340857, −4.29176289442390821036698541598, −3.61277740438840229681447723540, −3.49618065958962322963341484542, −2.72518744743844417971804520987, −1.66749032540359766610848917641, −0.988439049029826398924798149053, 0.988439049029826398924798149053, 1.66749032540359766610848917641, 2.72518744743844417971804520987, 3.49618065958962322963341484542, 3.61277740438840229681447723540, 4.29176289442390821036698541598, 5.18278653808789891360302340857, 5.39281692481958639495063470626, 5.95634878468840258125794611097, 6.58418622739578159912508579609, 7.22891664623198580869388782053, 7.50830134670769843729117769646, 8.045451793964213293582863010431, 8.594949072848139512290444008011, 8.814921481318144229906760633094

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