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Properties

Label 4-4212e2-1.1-c1e2-0-11
Degree $4$
Conductor $17740944$
Sign $1$
Analytic cond. $1131.17$
Root an. cond. $5.79939$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 13-s + 12·17-s + 4·19-s + 5·25-s − 6·29-s − 2·31-s + 4·37-s − 12·41-s + 4·43-s + 7·49-s − 12·53-s + 12·59-s − 2·61-s + 10·67-s − 24·71-s + 28·73-s − 8·79-s + 12·83-s + 2·91-s + 10·97-s + 18·101-s + 16·103-s + 24·107-s + 28·109-s + 6·113-s − 24·119-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.277·13-s + 2.91·17-s + 0.917·19-s + 25-s − 1.11·29-s − 0.359·31-s + 0.657·37-s − 1.87·41-s + 0.609·43-s + 49-s − 1.64·53-s + 1.56·59-s − 0.256·61-s + 1.22·67-s − 2.84·71-s + 3.27·73-s − 0.900·79-s + 1.31·83-s + 0.209·91-s + 1.01·97-s + 1.79·101-s + 1.57·103-s + 2.32·107-s + 2.68·109-s + 0.564·113-s − 2.20·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17740944\)    =    \(2^{4} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1131.17\)
Root analytic conductor: \(5.79939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17740944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.328328720\)
\(L(\frac12)\) \(\approx\) \(3.328328720\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_bh
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−8.614014174793246721113755705275, −8.143553346055720430942235596815, −7.69539199413123751589677082495, −7.58243384334465302098618819568, −7.14844492485374350441715973146, −6.93062644212712924489559958490, −6.22489766085349004328300136703, −6.05846308948895687089457856837, −5.58289516293877438294893616204, −5.36080263771802947363668214162, −4.82781622868573726056127004554, −4.65089872151306710121541746823, −3.69551268326919292542764562696, −3.59161748305652296674198671519, −3.12798107092266865858283211019, −3.04187594877793618101363243463, −2.08814678888175711037273808641, −1.79068614878330914646887036021, −0.824311461145586577499358771977, −0.73372551445957873793040119268, 0.73372551445957873793040119268, 0.824311461145586577499358771977, 1.79068614878330914646887036021, 2.08814678888175711037273808641, 3.04187594877793618101363243463, 3.12798107092266865858283211019, 3.59161748305652296674198671519, 3.69551268326919292542764562696, 4.65089872151306710121541746823, 4.82781622868573726056127004554, 5.36080263771802947363668214162, 5.58289516293877438294893616204, 6.05846308948895687089457856837, 6.22489766085349004328300136703, 6.93062644212712924489559958490, 7.14844492485374350441715973146, 7.58243384334465302098618819568, 7.69539199413123751589677082495, 8.143553346055720430942235596815, 8.614014174793246721113755705275

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