Properties

Label 4-913952-1.1-c1e2-0-0
Degree $4$
Conductor $913952$
Sign $1$
Analytic cond. $58.2743$
Root an. cond. $2.76292$
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 + 4-s − 2·5-s + 8-s − 6·9-s − 2·10-s + 16-s + 6·17-s − 6·18-s − 2·20-s − 7·25-s − 2·29-s + 32-s + 6·34-s − 6·36-s + 6·37-s − 2·40-s − 18·41-s + 12·45-s + 2·49-s − 7·50-s − 18·53-s − 2·58-s + 14·61-s + 64-s + 6·68-s − 6·72-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 2·9-s − 0.632·10-s + 1/4·16-s + 1.45·17-s − 1.41·18-s − 0.447·20-s − 7/5·25-s − 0.371·29-s + 0.176·32-s + 1.02·34-s − 36-s + 0.986·37-s − 0.316·40-s − 2.81·41-s + 1.78·45-s + 2/7·49-s − 0.989·50-s − 2.47·53-s − 0.262·58-s + 1.79·61-s + 1/8·64-s + 0.727·68-s − 0.707·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 913952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 913952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(913952\)    =    \(2^{5} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58.2743\)
Root analytic conductor: \(2.76292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 913952,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.452666500\)
\(L(\frac12)\) \(\approx\) \(1.452666500\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.002291181956618059512371438305, −7.911995312311956948638277141106, −7.41672588199918751908312168024, −6.56150790811441553407226286731, −6.50399725127484965649898426018, −5.72855680464161796505975732777, −5.49349708810384813454586687760, −5.13163321247773548329189218957, −4.50196070453734734936420698072, −3.79024118842185882317413842973, −3.41530945529539786727500876542, −3.16674674785360709298080624336, −2.40515532571205298049056395316, −1.72818916811260562211346322788, −0.48545352491889199332359964707, 0.48545352491889199332359964707, 1.72818916811260562211346322788, 2.40515532571205298049056395316, 3.16674674785360709298080624336, 3.41530945529539786727500876542, 3.79024118842185882317413842973, 4.50196070453734734936420698072, 5.13163321247773548329189218957, 5.49349708810384813454586687760, 5.72855680464161796505975732777, 6.50399725127484965649898426018, 6.56150790811441553407226286731, 7.41672588199918751908312168024, 7.911995312311956948638277141106, 8.002291181956618059512371438305

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