Properties

Label 4-390e2-1.1-c1e2-0-13
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  + 2-s + 3-s + 2·5-s + 6-s + 2·7-s − 8-s + 2·10-s + 5·11-s − 2·13-s + 2·14-s + 2·15-s − 16-s + 2·17-s + 2·19-s + 2·21-s + 5·22-s + 23-s − 24-s + 3·25-s − 2·26-s − 27-s − 5·29-s + 2·30-s − 22·31-s + 5·33-s + 2·34-s + 4·35-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.632·10-s + 1.50·11-s − 0.554·13-s + 0.534·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 1.06·22-s + 0.208·23-s − 0.204·24-s + 3/5·25-s − 0.392·26-s − 0.192·27-s − 0.928·29-s + 0.365·30-s − 3.95·31-s + 0.870·33-s + 0.342·34-s + 0.676·35-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

\(L(1)\) \(\approx\) \(3.652644331\)
\(L(\frac12)\) \(\approx\) \(3.652644331\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−11.59851873382212028174123408380, −11.18442299744367113597707158431, −10.52579127363265289146473027770, −10.36376533340799082646462021513, −9.432965150138429998936672765786, −9.123968642820724329699847085597, −9.044523223018925566985447037164, −8.584713050959618961038837565638, −7.44217447050888354884752365617, −7.37403208304333244737649204443, −7.07782553058567051413845670594, −5.93254172882572168601355129205, −5.65834962314051177795458970594, −5.47788980204384098368486761399, −4.58281741629197366423045951803, −3.85803129405172033292355299156, −3.75465106684680250691141445322, −2.69869632305814641377915641814, −2.05629244924098450802776565712, −1.31667554819934426472056041060, 1.31667554819934426472056041060, 2.05629244924098450802776565712, 2.69869632305814641377915641814, 3.75465106684680250691141445322, 3.85803129405172033292355299156, 4.58281741629197366423045951803, 5.47788980204384098368486761399, 5.65834962314051177795458970594, 5.93254172882572168601355129205, 7.07782553058567051413845670594, 7.37403208304333244737649204443, 7.44217447050888354884752365617, 8.584713050959618961038837565638, 9.044523223018925566985447037164, 9.123968642820724329699847085597, 9.432965150138429998936672765786, 10.36376533340799082646462021513, 10.52579127363265289146473027770, 11.18442299744367113597707158431, 11.59851873382212028174123408380

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