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Properties

Label 4-33e4-1.1-c1e2-0-13
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $75.6153$
Root an. cond. $2.94884$
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  − 4-s + 6·5-s − 3·16-s − 6·20-s + 12·23-s + 17·25-s + 8·31-s − 22·37-s − 2·49-s + 18·53-s + 12·59-s + 7·64-s − 4·67-s + 12·71-s − 18·80-s − 18·89-s − 12·92-s − 14·97-s − 17·100-s + 28·103-s + 42·113-s + 72·115-s − 8·124-s + 18·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.68·5-s − 3/4·16-s − 1.34·20-s + 2.50·23-s + 17/5·25-s + 1.43·31-s − 3.61·37-s − 2/7·49-s + 2.47·53-s + 1.56·59-s + 7/8·64-s − 0.488·67-s + 1.42·71-s − 2.01·80-s − 1.90·89-s − 1.25·92-s − 1.42·97-s − 1.69·100-s + 2.75·103-s + 3.95·113-s + 6.71·115-s − 0.718·124-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(75.6153\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.447865298\)
\(L(\frac12)\) \(\approx\) \(3.447865298\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.29.a_cd
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.41.a_db
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.73.a_du
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
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

−10.02086063647710924178450116043, −9.842325386386337692590707857480, −9.216354959181625752243332828814, −8.870987129926948099475686342112, −8.596938216441731266076463871288, −8.450993844397007973929666407794, −7.26847517935432064212987032117, −7.00607633369541907559955104744, −6.81046356117150327321636010509, −6.19728146879401851309670428527, −5.76558471103732495983973628130, −5.37454660274404579404395867177, −4.87171789207207126244139650640, −4.86693552311694469802699538494, −3.83786711455443175585011229911, −3.27007684045341957162730112055, −2.55800580814670307665353109157, −2.21393848240163495721722246576, −1.59063782793091227817997843551, −0.873365374176186409069496759274, 0.873365374176186409069496759274, 1.59063782793091227817997843551, 2.21393848240163495721722246576, 2.55800580814670307665353109157, 3.27007684045341957162730112055, 3.83786711455443175585011229911, 4.86693552311694469802699538494, 4.87171789207207126244139650640, 5.37454660274404579404395867177, 5.76558471103732495983973628130, 6.19728146879401851309670428527, 6.81046356117150327321636010509, 7.00607633369541907559955104744, 7.26847517935432064212987032117, 8.450993844397007973929666407794, 8.596938216441731266076463871288, 8.870987129926948099475686342112, 9.216354959181625752243332828814, 9.842325386386337692590707857480, 10.02086063647710924178450116043

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