Properties

Label 4-968e2-1.1-c1e2-0-8
Degree $4$
Conductor $937024$
Sign $1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 5·9-s + 8·13-s − 12·17-s + 17·25-s − 2·37-s + 30·45-s − 10·49-s − 12·53-s + 8·61-s − 48·65-s + 8·73-s + 16·81-s + 72·85-s − 18·89-s − 14·97-s − 36·101-s − 4·109-s − 30·113-s − 40·117-s − 18·125-s + ⋯
L(s)  = 1  − 2.68·5-s − 5/3·9-s + 2.21·13-s − 2.91·17-s + 17/5·25-s − 0.328·37-s + 4.47·45-s − 1.42·49-s − 1.64·53-s + 1.02·61-s − 5.95·65-s + 0.936·73-s + 16/9·81-s + 7.80·85-s − 1.90·89-s − 1.42·97-s − 3.58·101-s − 0.383·109-s − 2.82·113-s − 3.69·117-s − 1.60·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.67.a_fd
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.a_fy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.97.o_jj
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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

−8.106469985083330538380333325542, −7.26152152624268428704712414891, −6.82620843631036387433675479847, −6.49340012695586359351910248567, −6.07532856213092694592866222076, −5.43339609963471466451709551596, −4.79469791447770690232597181563, −4.33075028432463196258949288256, −3.82876273665516882094535074726, −3.68366753852883183133176875421, −3.00443596972753636527090013772, −2.47303941619863806873155456193, −1.36947441810401211657481497357, 0, 0, 1.36947441810401211657481497357, 2.47303941619863806873155456193, 3.00443596972753636527090013772, 3.68366753852883183133176875421, 3.82876273665516882094535074726, 4.33075028432463196258949288256, 4.79469791447770690232597181563, 5.43339609963471466451709551596, 6.07532856213092694592866222076, 6.49340012695586359351910248567, 6.82620843631036387433675479847, 7.26152152624268428704712414891, 8.106469985083330538380333325542

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