Properties

Label 4-3234e2-1.1-c1e2-0-6
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 4·8-s + 3·9-s − 4·10-s − 2·11-s − 6·12-s + 4·15-s + 5·16-s − 6·17-s + 6·18-s − 2·19-s − 6·20-s − 4·22-s + 4·23-s − 8·24-s − 2·25-s − 4·27-s + 8·30-s − 10·31-s + 6·32-s + 4·33-s − 12·34-s + 9·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s − 2/5·25-s − 0.769·27-s + 1.46·30-s − 1.79·31-s + 1.06·32-s + 0.696·33-s − 2.05·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3234} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−8.308395201887522467306379368116, −7.937319973966220905190776043721, −7.45554471439769903633781626731, −7.25495299417794870485325880886, −6.87453239831790004284162762187, −6.51425790032642295735257257158, −6.06269675782315864237222366731, −5.86755441476510750574776400246, −5.20488699111769931025085260878, −5.08140986852376064045951094602, −4.48584054206085685239316898852, −4.48494736581129919926470525921, −3.74614230975455923025774579111, −3.57322553777284998809167820847, −2.96250317694907553878966878241, −2.43139618248874823913260659253, −1.78668960341985262097848069562, −1.39274818428326106592629821662, 0, 0, 1.39274818428326106592629821662, 1.78668960341985262097848069562, 2.43139618248874823913260659253, 2.96250317694907553878966878241, 3.57322553777284998809167820847, 3.74614230975455923025774579111, 4.48494736581129919926470525921, 4.48584054206085685239316898852, 5.08140986852376064045951094602, 5.20488699111769931025085260878, 5.86755441476510750574776400246, 6.06269675782315864237222366731, 6.51425790032642295735257257158, 6.87453239831790004284162762187, 7.25495299417794870485325880886, 7.45554471439769903633781626731, 7.937319973966220905190776043721, 8.308395201887522467306379368116

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