↑

Properties

Label 4-2376e2-1.1-c1e2-0-10
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 9-s − 4·11-s − 2·15-s − 6·25-s + 27-s + 2·31-s − 4·33-s − 10·37-s − 2·45-s + 2·47-s − 2·49-s − 2·53-s + 8·55-s + 4·59-s + 6·67-s − 6·71-s − 6·75-s + 81-s + 16·89-s + 2·93-s − 97-s − 4·99-s + 31·103-s − 10·111-s + 4·113-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.516·15-s − 6/5·25-s + 0.192·27-s + 0.359·31-s − 0.696·33-s − 1.64·37-s − 0.298·45-s + 0.291·47-s − 2/7·49-s − 0.274·53-s + 1.07·55-s + 0.520·59-s + 0.733·67-s − 0.712·71-s − 0.692·75-s + 1/9·81-s + 1.69·89-s + 0.207·93-s − 0.101·97-s − 0.402·99-s + 3.05·103-s − 0.949·111-s + 0.376·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.c_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.13.a_ap
17$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.17.a_an
19$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.19.a_ap
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.29.a_ab
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.k_df
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.41.a_cd
43$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.43.a_at
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$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.c_du
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.59.ae_ec
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.61.a_cz
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.67.ag_ex
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.g_dy
73$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.73.a_bp
79$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \) 2.79.a_dd
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.89.aq_is
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.b_hk
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.23868121460037272703718688780, −6.84055525959904533677165625070, −6.15480849716659002368854673388, −5.94594686122187572258831846273, −5.35181241065617502258535489550, −4.89434811589607643749094219095, −4.65794667052443203443250269404, −3.96845652464220267484230023982, −3.68720249822342675749100585778, −3.24631539346860659286178538803, −2.78245392088727725568903182309, −2.10287108212953815668973419547, −1.82475016266474522224517264237, −0.76678525132794806699394201304, 0, 0.76678525132794806699394201304, 1.82475016266474522224517264237, 2.10287108212953815668973419547, 2.78245392088727725568903182309, 3.24631539346860659286178538803, 3.68720249822342675749100585778, 3.96845652464220267484230023982, 4.65794667052443203443250269404, 4.89434811589607643749094219095, 5.35181241065617502258535489550, 5.94594686122187572258831846273, 6.15480849716659002368854673388, 6.84055525959904533677165625070, 7.23868121460037272703718688780

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