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Properties

Label 4-3833280-1.1-c1e2-0-11
Degree $4$
Conductor $3833280$
Sign $1$
Analytic cond. $244.413$
Root an. cond. $3.95395$
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  + 2·3-s − 3·5-s + 3·9-s − 11-s − 6·15-s + 10·23-s + 8·25-s + 4·27-s + 8·31-s − 2·33-s + 8·37-s − 9·45-s + 2·47-s − 2·49-s + 10·53-s + 3·55-s + 12·59-s + 20·67-s + 20·69-s + 10·71-s + 16·75-s + 5·81-s − 20·89-s + 16·93-s + 8·97-s − 3·99-s + 12·103-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s + 9-s − 0.301·11-s − 1.54·15-s + 2.08·23-s + 8/5·25-s + 0.769·27-s + 1.43·31-s − 0.348·33-s + 1.31·37-s − 1.34·45-s + 0.291·47-s − 2/7·49-s + 1.37·53-s + 0.404·55-s + 1.56·59-s + 2.44·67-s + 2.40·69-s + 1.18·71-s + 1.84·75-s + 5/9·81-s − 2.11·89-s + 1.65·93-s + 0.812·97-s − 0.301·99-s + 1.18·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3833280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3833280\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(244.413\)
Root analytic conductor: \(3.95395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3833280,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.495540568\)
\(L(\frac12)\) \(\approx\) \(3.495540568\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
11$C_1$ \( 1 + T \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.ak_cs
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.31.ai_cw
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.41.a_acs
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ac_cs
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.ak_fa
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.am_fu
61$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.61.a_co
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.au_iw
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.ak_bu
73$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.73.a_ec
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.83.a_abm
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ai_eg
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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

−7.37893175832879987879852026341, −7.22540170333877253143240738374, −6.71915936912420246471238810072, −6.53694726224203153430996113255, −5.69425662619810295717087333904, −5.23151223481709996928687553010, −4.75769357200990591526144166009, −4.43760575839282960036075668257, −3.90892801953354139477602179624, −3.53784901404056268268087215533, −3.05940902768935328487213930985, −2.58418853524516582497904262427, −2.26264659570289265246965541428, −1.07916952260744583128544651807, −0.77942797257599198400657334664, 0.77942797257599198400657334664, 1.07916952260744583128544651807, 2.26264659570289265246965541428, 2.58418853524516582497904262427, 3.05940902768935328487213930985, 3.53784901404056268268087215533, 3.90892801953354139477602179624, 4.43760575839282960036075668257, 4.75769357200990591526144166009, 5.23151223481709996928687553010, 5.69425662619810295717087333904, 6.53694726224203153430996113255, 6.71915936912420246471238810072, 7.22540170333877253143240738374, 7.37893175832879987879852026341

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