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Properties

Label 4-264e2-1.1-c1e2-0-17
Degree $4$
Conductor $69696$
Sign $-1$
Analytic cond. $4.44387$
Root an. cond. $1.45191$
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  − 3·3-s − 4·7-s + 6·9-s + 8·19-s + 12·21-s − 25-s − 9·27-s − 14·31-s − 2·37-s + 12·43-s − 2·49-s − 24·57-s + 8·61-s − 24·63-s − 10·67-s + 32·73-s + 3·75-s + 4·79-s + 9·81-s + 42·93-s − 14·97-s − 32·103-s − 28·109-s + 6·111-s + 121-s + 127-s − 36·129-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.51·7-s + 2·9-s + 1.83·19-s + 2.61·21-s − 1/5·25-s − 1.73·27-s − 2.51·31-s − 0.328·37-s + 1.82·43-s − 2/7·49-s − 3.17·57-s + 1.02·61-s − 3.02·63-s − 1.22·67-s + 3.74·73-s + 0.346·75-s + 0.450·79-s + 81-s + 4.35·93-s − 1.42·97-s − 3.15·103-s − 2.68·109-s + 0.569·111-s + 1/11·121-s + 0.0887·127-s − 3.16·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(4.44387\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 69696,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.a_co
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.59.a_en
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.67.k_gd
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.73.abg_pm
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
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

−9.698101458189890687450989475491, −9.401670233685537405973889044671, −8.798126319316295192274656678751, −7.62736396087440842811173385580, −7.57356519540538183281509096087, −6.66076665440645270819143387724, −6.61017530637467554226176574176, −5.84888418287605084778278372505, −5.26718062023912132586056594917, −5.18716225391118521175463046445, −3.92131200412791242288608029354, −3.66549703507533128595500156466, −2.64863885426300166607291973733, −1.27435416840477191188067329864, 0, 1.27435416840477191188067329864, 2.64863885426300166607291973733, 3.66549703507533128595500156466, 3.92131200412791242288608029354, 5.18716225391118521175463046445, 5.26718062023912132586056594917, 5.84888418287605084778278372505, 6.61017530637467554226176574176, 6.66076665440645270819143387724, 7.57356519540538183281509096087, 7.62736396087440842811173385580, 8.798126319316295192274656678751, 9.401670233685537405973889044671, 9.698101458189890687450989475491

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