Properties

Label 4-390e2-1.1-c1e2-0-8
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
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 + 4·11-s + 16-s − 4·19-s − 4·20-s + 11·25-s + 4·29-s − 8·31-s + 36-s − 12·41-s − 4·44-s − 4·45-s − 2·49-s + 16·55-s + 28·59-s + 20·61-s − 64-s + 16·71-s + 4·76-s + 16·79-s + 4·80-s + 81-s + 36·89-s − 16·95-s − 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 0.917·19-s − 0.894·20-s + 11/5·25-s + 0.742·29-s − 1.43·31-s + 1/6·36-s − 1.87·41-s − 0.603·44-s − 0.596·45-s − 2/7·49-s + 2.15·55-s + 3.64·59-s + 2.56·61-s − 1/8·64-s + 1.89·71-s + 0.458·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 3.81·89-s − 1.64·95-s − 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{390} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106859932\)
\(L(\frac12)\) \(\approx\) \(2.106859932\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−11.70541008195990028035299899017, −10.95030909764592178474522169338, −10.44950589932283994661408709602, −10.23845136228507693970389687210, −9.475364833362792739908843742432, −9.453993206689234124581425350749, −8.927157072855751300511983482411, −8.323560979338947872027091728120, −8.199621978548240832095296968422, −7.02555179057765223695230575436, −6.59800822985621947604709322250, −6.52567964516979694630917506526, −5.69097944702842112330053025984, −5.19510855767203085831627464049, −4.99600095108582412245908631971, −3.78286145293792027194094329732, −3.72794716259632351621374002454, −2.44101128541060251824225185472, −2.04586530366640465688845444023, −1.06042640914004630859042630322, 1.06042640914004630859042630322, 2.04586530366640465688845444023, 2.44101128541060251824225185472, 3.72794716259632351621374002454, 3.78286145293792027194094329732, 4.99600095108582412245908631971, 5.19510855767203085831627464049, 5.69097944702842112330053025984, 6.52567964516979694630917506526, 6.59800822985621947604709322250, 7.02555179057765223695230575436, 8.199621978548240832095296968422, 8.323560979338947872027091728120, 8.927157072855751300511983482411, 9.453993206689234124581425350749, 9.475364833362792739908843742432, 10.23845136228507693970389687210, 10.44950589932283994661408709602, 10.95030909764592178474522169338, 11.70541008195990028035299899017

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