Properties

Label 4-3675e2-1.1-c1e2-0-8
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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  + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 4·8-s + 3·9-s − 2·11-s − 4·12-s + 8·13-s + 8·16-s + 10·17-s + 6·18-s − 2·19-s − 4·22-s + 6·23-s − 8·24-s + 16·26-s − 4·27-s + 2·29-s − 6·31-s + 8·32-s + 4·33-s + 20·34-s + 6·36-s − 4·37-s − 4·38-s − 16·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 1.41·8-s + 9-s − 0.603·11-s − 1.15·12-s + 2.21·13-s + 2·16-s + 2.42·17-s + 1.41·18-s − 0.458·19-s − 0.852·22-s + 1.25·23-s − 1.63·24-s + 3.13·26-s − 0.769·27-s + 0.371·29-s − 1.07·31-s + 1.41·32-s + 0.696·33-s + 3.42·34-s + 36-s − 0.657·37-s − 0.648·38-s − 2.56·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3675} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.396109465\)
\(L(\frac12)\) \(\approx\) \(6.396109465\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 12 T + 95 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−8.634600473453895480720660911993, −8.190415223690661105187529285367, −7.78346763124498357013353427450, −7.43940302657860016755960654737, −7.33301485803695745849939701754, −6.57573638910107396292674728378, −6.21887629193999175199005858561, −6.14920323733806161412877523239, −5.50603161033232513432509219587, −5.37673462065514077764130003734, −4.91455032616381280109438159151, −4.82842337953180302293122281503, −3.95251876895028060613112539802, −3.86775738177741210899360661204, −3.36415960026344400067495415761, −3.17072792592381130922122229488, −2.30177497798395261531748571307, −1.53609626877493731914642008821, −1.26088179737818998207435618996, −0.72311973202418744922349492473, 0.72311973202418744922349492473, 1.26088179737818998207435618996, 1.53609626877493731914642008821, 2.30177497798395261531748571307, 3.17072792592381130922122229488, 3.36415960026344400067495415761, 3.86775738177741210899360661204, 3.95251876895028060613112539802, 4.82842337953180302293122281503, 4.91455032616381280109438159151, 5.37673462065514077764130003734, 5.50603161033232513432509219587, 6.14920323733806161412877523239, 6.21887629193999175199005858561, 6.57573638910107396292674728378, 7.33301485803695745849939701754, 7.43940302657860016755960654737, 7.78346763124498357013353427450, 8.190415223690661105187529285367, 8.634600473453895480720660911993

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