Properties

Label 4-930e2-1.1-c1e2-0-4
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  − 4-s + 4·5-s − 9-s − 12·11-s + 16-s − 16·19-s − 4·20-s + 11·25-s + 8·29-s − 2·31-s + 36-s + 4·41-s + 12·44-s − 4·45-s + 10·49-s − 48·55-s − 12·59-s − 28·61-s − 64-s + 32·71-s + 16·76-s + 4·80-s + 81-s − 20·89-s − 64·95-s + 12·99-s − 11·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s − 3.61·11-s + 1/4·16-s − 3.67·19-s − 0.894·20-s + 11/5·25-s + 1.48·29-s − 0.359·31-s + 1/6·36-s + 0.624·41-s + 1.80·44-s − 0.596·45-s + 10/7·49-s − 6.47·55-s − 1.56·59-s − 3.58·61-s − 1/8·64-s + 3.79·71-s + 1.83·76-s + 0.447·80-s + 1/9·81-s − 2.11·89-s − 6.56·95-s + 1.20·99-s − 1.09·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9353984850\)
\(L(\frac12)\) \(\approx\) \(0.9353984850\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.51562922229609008700824318812, −9.920692181020961282739520091759, −9.458820960308478406152581506333, −9.114882570523260308065727861260, −8.423286872269524714293322793995, −8.252289531777121079229109701580, −8.082992993984181268186689906233, −7.32187816876209936881109139987, −6.78361127894603513312779486550, −6.21364179357907683663796927527, −5.92531900103016535393686587314, −5.53786055272347795924294184933, −5.09057196258572380238194577904, −4.47727261522209926947945209115, −4.45309952440051840068136497396, −3.07971093944410303561855635872, −2.77623432088683007099835644420, −2.19071424827379465570161767971, −1.97375645400248099072466309148, −0.41938997965137985629949346235, 0.41938997965137985629949346235, 1.97375645400248099072466309148, 2.19071424827379465570161767971, 2.77623432088683007099835644420, 3.07971093944410303561855635872, 4.45309952440051840068136497396, 4.47727261522209926947945209115, 5.09057196258572380238194577904, 5.53786055272347795924294184933, 5.92531900103016535393686587314, 6.21364179357907683663796927527, 6.78361127894603513312779486550, 7.32187816876209936881109139987, 8.082992993984181268186689906233, 8.252289531777121079229109701580, 8.423286872269524714293322793995, 9.114882570523260308065727861260, 9.458820960308478406152581506333, 9.920692181020961282739520091759, 10.51562922229609008700824318812

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