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Properties

Label 4-1769472-1.1-c1e2-0-57
Degree $4$
Conductor $1769472$
Sign $-1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 4·11-s − 12·17-s + 4·19-s − 2·25-s + 27-s + 4·33-s + 12·41-s − 4·43-s + 6·49-s − 12·51-s + 4·57-s − 24·59-s − 16·67-s − 4·73-s − 2·75-s + 81-s − 12·83-s − 8·89-s + 4·97-s + 4·99-s + 16·107-s + 8·113-s − 6·121-s + 12·123-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 2.91·17-s + 0.917·19-s − 2/5·25-s + 0.192·27-s + 0.696·33-s + 1.87·41-s − 0.609·43-s + 6/7·49-s − 1.68·51-s + 0.529·57-s − 3.12·59-s − 1.95·67-s − 0.468·73-s − 0.230·75-s + 1/9·81-s − 1.31·83-s − 0.847·89-s + 0.406·97-s + 0.402·99-s + 1.54·107-s + 0.752·113-s − 0.545·121-s + 1.08·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.19.ae_bm
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.29.a_ba
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.am_dy
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.e_cc
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.61.a_bi
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.71.a_o
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.e_di
79$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.79.a_ba
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.m_dy
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.i_gc
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ae_ha
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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

−7.60549684726132289974795529255, −7.20437267937941753564790337253, −6.76788469853695283773722321208, −6.37834427559823210863863483134, −6.00750584155276541902078555310, −5.53655211171527223145402050161, −4.67705120596326002280604459280, −4.38057509860893759445611584853, −4.23673666115774008063653795967, −3.47465177703265710800817322291, −2.96296830478488251517687278855, −2.40485389266313143507580177090, −1.81778011840556996777631218205, −1.21596478224321659078650931970, 0, 1.21596478224321659078650931970, 1.81778011840556996777631218205, 2.40485389266313143507580177090, 2.96296830478488251517687278855, 3.47465177703265710800817322291, 4.23673666115774008063653795967, 4.38057509860893759445611584853, 4.67705120596326002280604459280, 5.53655211171527223145402050161, 6.00750584155276541902078555310, 6.37834427559823210863863483134, 6.76788469853695283773722321208, 7.20437267937941753564790337253, 7.60549684726132289974795529255

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