Properties

Label 4-7098e2-1.1-c1e2-0-1
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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 + 3·4-s − 3·5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 6·10-s + 7·11-s − 6·12-s + 4·14-s + 6·15-s + 5·16-s + 17-s − 6·18-s − 19-s − 9·20-s + 4·21-s − 14·22-s + 23-s + 8·24-s + 25-s − 4·27-s − 6·28-s − 29-s − 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.89·10-s + 2.11·11-s − 1.73·12-s + 1.06·14-s + 1.54·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s − 0.229·19-s − 2.01·20-s + 0.872·21-s − 2.98·22-s + 0.208·23-s + 1.63·24-s + 1/5·25-s − 0.769·27-s − 1.13·28-s − 0.185·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6722407694\)
\(L(\frac12)\) \(\approx\) \(0.6722407694\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 17 T + 154 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 19 T + 232 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.950582076466940698145219677834, −7.87473873718952103891309868393, −7.24833967352261023685898319759, −7.21940195867006375357813191464, −6.73188132240621946240011159912, −6.54970724765599673234416140709, −6.08346639879335781299519797261, −5.98809613404291535315695424318, −5.36267124498753473992275493470, −5.07760720067687761371554941761, −4.22998135057417449139168524993, −4.19617924627748042340852354755, −3.72381044853658398152580713167, −3.56513794513188689145523150626, −2.89827504331595590276262707192, −2.31638597677632616093771633069, −1.82597683556651211561604258599, −1.18989435134740020733538435282, −0.75371095167969600174767616774, −0.44180928026596389021966686337, 0.44180928026596389021966686337, 0.75371095167969600174767616774, 1.18989435134740020733538435282, 1.82597683556651211561604258599, 2.31638597677632616093771633069, 2.89827504331595590276262707192, 3.56513794513188689145523150626, 3.72381044853658398152580713167, 4.19617924627748042340852354755, 4.22998135057417449139168524993, 5.07760720067687761371554941761, 5.36267124498753473992275493470, 5.98809613404291535315695424318, 6.08346639879335781299519797261, 6.54970724765599673234416140709, 6.73188132240621946240011159912, 7.21940195867006375357813191464, 7.24833967352261023685898319759, 7.87473873718952103891309868393, 7.950582076466940698145219677834

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