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Properties

Label 4-189e2-1.1-c1e2-0-16
Degree $4$
Conductor $35721$
Sign $-1$
Analytic cond. $2.27760$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·13-s − 4·16-s − 16·19-s − 9·25-s + 12·31-s − 6·37-s + 22·43-s + 3·49-s + 8·61-s − 16·67-s + 12·73-s − 2·79-s + 8·91-s + 24·97-s + 4·103-s − 18·109-s + 8·112-s − 6·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.10·13-s − 16-s − 3.67·19-s − 9/5·25-s + 2.15·31-s − 0.986·37-s + 3.35·43-s + 3/7·49-s + 1.02·61-s − 1.95·67-s + 1.40·73-s − 0.225·79-s + 0.838·91-s + 2.43·97-s + 0.394·103-s − 1.72·109-s + 0.755·112-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35721\)    =    \(3^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(2.27760\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 35721,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.41.a_dd
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.59.a_aed
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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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

−10.17243898255102214941018911651, −9.606127837407754601596009449174, −9.039863529492895542714101940401, −8.650153087392553143775802165667, −8.011472352484350129239935128289, −7.43939934466159987675241180651, −6.75587037635177377501803178294, −6.25471251304940803802164013919, −5.97786895192084747528753482689, −4.86590526026148497681960970818, −4.28481394661801806092561213309, −3.90517995805379232478125425669, −2.42236155159716987191201970387, −2.34130595001443635884001685756, 0, 2.34130595001443635884001685756, 2.42236155159716987191201970387, 3.90517995805379232478125425669, 4.28481394661801806092561213309, 4.86590526026148497681960970818, 5.97786895192084747528753482689, 6.25471251304940803802164013919, 6.75587037635177377501803178294, 7.43939934466159987675241180651, 8.011472352484350129239935128289, 8.650153087392553143775802165667, 9.039863529492895542714101940401, 9.606127837407754601596009449174, 10.17243898255102214941018911651

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