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Properties

Label 4-1170e2-1.1-c1e2-0-14
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 + 7-s + 8-s − 2·10-s + 3·11-s + 5·13-s − 14-s − 16-s − 6·17-s − 5·19-s − 3·22-s + 3·25-s − 5·26-s − 8·31-s + 6·34-s + 2·35-s − 11·37-s + 5·38-s + 2·40-s + 6·41-s − 2·43-s + 6·47-s + 7·49-s − 3·50-s + 18·53-s + 6·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.904·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 1.14·19-s − 0.639·22-s + 3/5·25-s − 0.980·26-s − 1.43·31-s + 1.02·34-s + 0.338·35-s − 1.80·37-s + 0.811·38-s + 0.316·40-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 49-s − 0.424·50-s + 2.47·53-s + 0.809·55-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

\(L(1)\) \(\approx\) \(1.787461049\)
\(L(\frac12)\) \(\approx\) \(1.787461049\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.aq_hh
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_ai
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−9.916732566382925640226647820211, −9.422618477591754766479554891984, −9.045298273466182787870796596005, −8.847369243257180812431032660275, −8.493017871259422587394446632965, −8.266274093691874252632780008401, −7.43251362007141889403805249860, −7.09914997887592321606539854201, −6.56518857049837417056191977900, −6.46359244763172329999033479885, −5.67681541707511079774628834369, −5.55378504344832800887056352650, −4.84311983640726876920877448199, −4.26162298282323903470022789397, −3.84637623562808244760566236234, −3.49767744318245298164583121366, −2.24620458241471837175061669364, −2.19629460654248772239046829982, −1.46638571394123082379151316002, −0.68701130109115560134573676209, 0.68701130109115560134573676209, 1.46638571394123082379151316002, 2.19629460654248772239046829982, 2.24620458241471837175061669364, 3.49767744318245298164583121366, 3.84637623562808244760566236234, 4.26162298282323903470022789397, 4.84311983640726876920877448199, 5.55378504344832800887056352650, 5.67681541707511079774628834369, 6.46359244763172329999033479885, 6.56518857049837417056191977900, 7.09914997887592321606539854201, 7.43251362007141889403805249860, 8.266274093691874252632780008401, 8.493017871259422587394446632965, 8.847369243257180812431032660275, 9.045298273466182787870796596005, 9.422618477591754766479554891984, 9.916732566382925640226647820211

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