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Properties

Label 4-1752192-1.1-c1e2-0-6
Degree $4$
Conductor $1752192$
Sign $-1$
Analytic cond. $111.721$
Root an. cond. $3.25112$
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-s + 4-s + 8-s + 8·11-s + 16-s − 12·19-s + 8·22-s − 6·25-s + 32-s − 12·38-s − 12·41-s − 16·43-s + 8·44-s − 10·49-s − 6·50-s − 8·59-s + 64-s − 4·67-s + 28·73-s − 12·76-s − 12·82-s + 24·83-s − 16·86-s + 8·88-s + 12·89-s − 20·97-s − 10·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2.41·11-s + 1/4·16-s − 2.75·19-s + 1.70·22-s − 6/5·25-s + 0.176·32-s − 1.94·38-s − 1.87·41-s − 2.43·43-s + 1.20·44-s − 1.42·49-s − 0.848·50-s − 1.04·59-s + 1/8·64-s − 0.488·67-s + 3.27·73-s − 1.37·76-s − 1.32·82-s + 2.63·83-s − 1.72·86-s + 0.852·88-s + 1.27·89-s − 2.03·97-s − 1.01·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1752192 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1752192 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1752192\)    =    \(2^{7} \cdot 3^{4} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(111.721\)
Root analytic conductor: \(3.25112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1752192,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3 \( 1 \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.55343001357246297047779257346, −6.90620193553651768805380009960, −6.49638165287752346862203973768, −6.33856513547641234727266817168, −6.27277474889818422879179188201, −5.35933717968472431043686479964, −4.80645914681781210957697738257, −4.56665368749417880412544496052, −3.93890393883931852487830519645, −3.53791924823882088462258917674, −3.38762912518254017040905929558, −2.09554863734803568973997193711, −2.01663812390586556648231292076, −1.32459522720519948691308533254, 0, 1.32459522720519948691308533254, 2.01663812390586556648231292076, 2.09554863734803568973997193711, 3.38762912518254017040905929558, 3.53791924823882088462258917674, 3.93890393883931852487830519645, 4.56665368749417880412544496052, 4.80645914681781210957697738257, 5.35933717968472431043686479964, 6.27277474889818422879179188201, 6.33856513547641234727266817168, 6.49638165287752346862203973768, 6.90620193553651768805380009960, 7.55343001357246297047779257346

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