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Properties

Label 4-148e2-1.1-c1e2-0-6
Degree $4$
Conductor $21904$
Sign $1$
Analytic cond. $1.39661$
Root an. cond. $1.08709$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s − 3·9-s + 6·11-s + 4·21-s − 2·25-s − 14·27-s + 12·33-s − 10·37-s − 6·41-s + 18·47-s − 11·49-s − 6·53-s − 6·63-s + 8·67-s + 30·71-s + 2·73-s − 4·75-s + 12·77-s − 4·81-s − 18·83-s − 18·99-s + 18·101-s − 24·107-s − 20·111-s + 5·121-s − 12·123-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s − 9-s + 1.80·11-s + 0.872·21-s − 2/5·25-s − 2.69·27-s + 2.08·33-s − 1.64·37-s − 0.937·41-s + 2.62·47-s − 1.57·49-s − 0.824·53-s − 0.755·63-s + 0.977·67-s + 3.56·71-s + 0.234·73-s − 0.461·75-s + 1.36·77-s − 4/9·81-s − 1.97·83-s − 1.80·99-s + 1.79·101-s − 2.32·107-s − 1.89·111-s + 5/11·121-s − 1.08·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21904\)    =    \(2^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1.39661\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.716565710\)
\(L(\frac12)\) \(\approx\) \(1.716565710\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.17.a_aw
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.47.as_gt
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.71.abe_od
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.79.a_afq
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.89.a_acs
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−13.64119659592547014020348017621, −12.79260774949218116253447878423, −12.18076313272930523904805059494, −11.81453097122860723487542585254, −11.22205287137772566045163141236, −11.06188050714278931130835766692, −10.11840936861237292359097485507, −9.488910910367242339137864165231, −9.028923537585328699644848719058, −8.719262710601381121390326388858, −8.147251333294134589241289589943, −7.78258368612211783764055171838, −6.87956446346065506508968227144, −6.40118300747641518774513522912, −5.57914016287342457069619334456, −5.01139584170422053817190847606, −3.86829114593658313986535442433, −3.60339125412276415649951407652, −2.57626542035697030839244283263, −1.68720682565910526933776948835, 1.68720682565910526933776948835, 2.57626542035697030839244283263, 3.60339125412276415649951407652, 3.86829114593658313986535442433, 5.01139584170422053817190847606, 5.57914016287342457069619334456, 6.40118300747641518774513522912, 6.87956446346065506508968227144, 7.78258368612211783764055171838, 8.147251333294134589241289589943, 8.719262710601381121390326388858, 9.028923537585328699644848719058, 9.488910910367242339137864165231, 10.11840936861237292359097485507, 11.06188050714278931130835766692, 11.22205287137772566045163141236, 11.81453097122860723487542585254, 12.18076313272930523904805059494, 12.79260774949218116253447878423, 13.64119659592547014020348017621

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