Properties

Label 4-385e2-1.1-c1e2-0-0
Degree $4$
Conductor $148225$
Sign $1$
Analytic cond. $9.45095$
Root an. cond. $1.75335$
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  + 2·3-s − 4·4-s − 2·5-s − 3·9-s − 3·11-s − 8·12-s − 4·15-s + 12·16-s + 8·20-s − 12·23-s + 3·25-s − 14·27-s − 8·31-s − 6·33-s + 12·36-s + 4·37-s + 12·44-s + 6·45-s + 18·47-s + 24·48-s + 49-s + 24·53-s + 6·55-s + 16·60-s − 32·64-s − 8·67-s − 24·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 0.894·5-s − 9-s − 0.904·11-s − 2.30·12-s − 1.03·15-s + 3·16-s + 1.78·20-s − 2.50·23-s + 3/5·25-s − 2.69·27-s − 1.43·31-s − 1.04·33-s + 2·36-s + 0.657·37-s + 1.80·44-s + 0.894·45-s + 2.62·47-s + 3.46·48-s + 1/7·49-s + 3.29·53-s + 0.809·55-s + 2.06·60-s − 4·64-s − 0.977·67-s − 2.88·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(148225\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(9.45095\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{148225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 148225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4673276304\)
\(L(\frac12)\) \(\approx\) \(0.4673276304\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.112268205680581973965662028240, −8.679498853252053444462430241617, −8.504764710422860488143194115790, −7.980947296320422939728903060714, −7.67950449335459092353880992506, −7.28726628142540726038141332638, −5.98086532326102961421297628547, −5.49671842174989089979243313601, −5.47907564040983295563221596641, −4.26526382517520030491171627830, −4.09257990991624762122042789987, −3.61689003045514298236484607491, −2.90308901018164947604047652054, −2.20945739428189555232801439935, −0.42426621729141121803731134362, 0.42426621729141121803731134362, 2.20945739428189555232801439935, 2.90308901018164947604047652054, 3.61689003045514298236484607491, 4.09257990991624762122042789987, 4.26526382517520030491171627830, 5.47907564040983295563221596641, 5.49671842174989089979243313601, 5.98086532326102961421297628547, 7.28726628142540726038141332638, 7.67950449335459092353880992506, 7.980947296320422939728903060714, 8.504764710422860488143194115790, 8.679498853252053444462430241617, 9.112268205680581973965662028240

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