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Properties

Label 4-1170e2-1.1-c1e2-0-20
Degree $4$
Conductor $1368900$
Sign $1$
Analytic cond. $87.2822$
Root an. cond. $3.05654$
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-s + 2·5-s − 2·7-s + 8-s − 2·10-s − 3·11-s + 2·13-s + 2·14-s − 16-s + 6·17-s − 2·19-s + 3·22-s + 3·23-s + 3·25-s − 2·26-s + 3·29-s + 10·31-s − 6·34-s − 4·35-s + 7·37-s + 2·38-s + 2·40-s + 6·41-s + 43-s − 3·46-s + 6·47-s + 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.894·5-s − 0.755·7-s + 0.353·8-s − 0.632·10-s − 0.904·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.639·22-s + 0.625·23-s + 3/5·25-s − 0.392·26-s + 0.557·29-s + 1.79·31-s − 1.02·34-s − 0.676·35-s + 1.15·37-s + 0.324·38-s + 0.316·40-s + 0.937·41-s + 0.152·43-s − 0.442·46-s + 0.875·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1368900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(87.2822\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1368900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.759956498\)
\(L(\frac12)\) \(\approx\) \(1.759956498\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2^2$ \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_cv
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.79.ak_hb
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_jb
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−9.898045651769643600425840887257, −9.702932306045277440553716292731, −9.131266351065574372980335278591, −8.891070491682525532386658414183, −8.396685719652457620059457196034, −8.027689691280649030350978385673, −7.43220661310585432294035941412, −7.38393643800771498406094090820, −6.43301638102232684230158465435, −6.37696470863339451746065675156, −5.74666410942358355966111509415, −5.56679264005520412935511674874, −4.75035014705503777374577663516, −4.55150920627240293930766179360, −3.69954522738404830254584846954, −3.22900913174491688577637999713, −2.53773791588137724588296101243, −2.32901142162861317005420996765, −1.13259063017847461427288780275, −0.791444958194323520660811044946, 0.791444958194323520660811044946, 1.13259063017847461427288780275, 2.32901142162861317005420996765, 2.53773791588137724588296101243, 3.22900913174491688577637999713, 3.69954522738404830254584846954, 4.55150920627240293930766179360, 4.75035014705503777374577663516, 5.56679264005520412935511674874, 5.74666410942358355966111509415, 6.37696470863339451746065675156, 6.43301638102232684230158465435, 7.38393643800771498406094090820, 7.43220661310585432294035941412, 8.027689691280649030350978385673, 8.396685719652457620059457196034, 8.891070491682525532386658414183, 9.131266351065574372980335278591, 9.702932306045277440553716292731, 9.898045651769643600425840887257

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