Properties

Label 4-148e2-1.1-c0e2-0-0
Degree $4$
Conductor $21904$
Sign $1$
Analytic cond. $0.00545553$
Root an. cond. $0.271774$
Motivic weight $0$
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 + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯
L(s)  = 1  − 2-s + 5-s + 8-s − 9-s − 10-s − 2·13-s − 16-s + 17-s + 18-s + 25-s + 2·26-s − 2·29-s − 34-s − 37-s + 40-s + 41-s − 45-s − 49-s − 50-s − 2·53-s + 2·58-s + 61-s + 64-s − 2·65-s − 72-s + 4·73-s + 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21904\)    =    \(2^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.00545553\)
Root analytic conductor: \(0.271774\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21904,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2597781081\)
\(L(\frac12)\) \(\approx\) \(0.2597781081\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
37$C_2$ \( 1 + T + T^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$ \( ( 1 - T )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−13.85653146219689087176981720472, −12.75470100440307614342689893959, −12.71611033927127344535869253417, −12.10175890340696933277954386344, −11.21638905910060143805761389051, −11.03020978873252149269118182461, −10.32016329323130379036524398995, −9.742048422211040471347849514437, −9.426467174471067737455659293965, −9.266654306864399411552743721958, −8.287180582741234828554201407768, −7.962051909607678993465956431210, −7.32941209753467813664766886872, −6.77605692322991098383087074077, −5.91882122147529498271655254947, −5.11379352929463618926966154777, −5.07555161507183373009572892502, −3.75374885505549034519191665355, −2.70594643519740103407472915111, −1.86391576307011732057411614853, 1.86391576307011732057411614853, 2.70594643519740103407472915111, 3.75374885505549034519191665355, 5.07555161507183373009572892502, 5.11379352929463618926966154777, 5.91882122147529498271655254947, 6.77605692322991098383087074077, 7.32941209753467813664766886872, 7.962051909607678993465956431210, 8.287180582741234828554201407768, 9.266654306864399411552743721958, 9.426467174471067737455659293965, 9.742048422211040471347849514437, 10.32016329323130379036524398995, 11.03020978873252149269118182461, 11.21638905910060143805761389051, 12.10175890340696933277954386344, 12.71611033927127344535869253417, 12.75470100440307614342689893959, 13.85653146219689087176981720472

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