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Properties

Label 4-1170e2-1.1-c1e2-0-19
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 5-s − 3·6-s − 2·7-s + 8-s + 6·9-s − 10-s − 11-s + 13-s + 2·14-s + 3·15-s − 16-s + 10·17-s − 6·18-s + 6·19-s − 6·21-s + 22-s + 3·24-s − 26-s + 9·27-s − 8·29-s − 3·30-s − 6·31-s − 3·33-s − 10·34-s − 2·35-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 0.447·5-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.534·14-s + 0.774·15-s − 1/4·16-s + 2.42·17-s − 1.41·18-s + 1.37·19-s − 1.30·21-s + 0.213·22-s + 0.612·24-s − 0.196·26-s + 1.73·27-s − 1.48·29-s − 0.547·30-s − 1.07·31-s − 0.522·33-s − 1.71·34-s − 0.338·35-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

\(L(1)\) \(\approx\) \(2.961776133\)
\(L(\frac12)\) \(\approx\) \(2.961776133\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bj
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_f
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.41.h_i
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.f_as
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dt
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.59.af_abi
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.67.ap_gc
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−10.04813052320260542104036511816, −9.542733364060192697479540794926, −9.050189613719278429468195452912, −8.728944802199760953130258409130, −8.705762549498479129573308356160, −7.88181439842747759055738872485, −7.51040352647099975014178989990, −7.33639120206070033381945123611, −7.13771498889159458550506118884, −6.23337387612351573109090487188, −5.77299585749420895778709015082, −5.22731937787428209721603290473, −5.06591458243817935201487163011, −3.86820911471094054670068148027, −3.70789082175443550073615353591, −3.19682486438564950890669974157, −2.98278483395025105590938094060, −1.86347692023748648719539082093, −1.76990634322557225197324702088, −0.77997820214502987274995757428, 0.77997820214502987274995757428, 1.76990634322557225197324702088, 1.86347692023748648719539082093, 2.98278483395025105590938094060, 3.19682486438564950890669974157, 3.70789082175443550073615353591, 3.86820911471094054670068148027, 5.06591458243817935201487163011, 5.22731937787428209721603290473, 5.77299585749420895778709015082, 6.23337387612351573109090487188, 7.13771498889159458550506118884, 7.33639120206070033381945123611, 7.51040352647099975014178989990, 7.88181439842747759055738872485, 8.705762549498479129573308356160, 8.728944802199760953130258409130, 9.050189613719278429468195452912, 9.542733364060192697479540794926, 10.04813052320260542104036511816

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