Properties

Label 4-33e4-1.1-c1e2-0-4
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $75.6153$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 2·5-s + 9-s + 12-s − 2·15-s − 3·16-s − 2·20-s − 11·23-s − 7·25-s + 27-s + 9·31-s + 36-s + 7·37-s − 2·45-s − 5·47-s − 3·48-s − 3·49-s − 6·53-s + 12·59-s − 2·60-s − 7·64-s + 24·67-s − 11·69-s − 4·71-s − 7·75-s + 6·80-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.894·5-s + 1/3·9-s + 0.288·12-s − 0.516·15-s − 3/4·16-s − 0.447·20-s − 2.29·23-s − 7/5·25-s + 0.192·27-s + 1.61·31-s + 1/6·36-s + 1.15·37-s − 0.298·45-s − 0.729·47-s − 0.433·48-s − 3/7·49-s − 0.824·53-s + 1.56·59-s − 0.258·60-s − 7/8·64-s + 2.93·67-s − 1.32·69-s − 0.474·71-s − 0.808·75-s + 0.670·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(75.6153\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.718875712\)
\(L(\frac12)\) \(\approx\) \(1.718875712\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 - T \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.942445275560774798604858549970, −7.81063907450705581096185251312, −7.29350762031471057748717102109, −6.68731653988993139741599168248, −6.32192236550450694976045486530, −6.05782356618264171710568495596, −5.34297086174204089539067431747, −4.73007373728730813936028654647, −4.25884488338889311249747851346, −3.86415008634793683968710904496, −3.50308751740273937443002527806, −2.68087772395802841862844943328, −2.24010662734099454406772102327, −1.73757767197656676707471694303, −0.54227682016181442004940467316, 0.54227682016181442004940467316, 1.73757767197656676707471694303, 2.24010662734099454406772102327, 2.68087772395802841862844943328, 3.50308751740273937443002527806, 3.86415008634793683968710904496, 4.25884488338889311249747851346, 4.73007373728730813936028654647, 5.34297086174204089539067431747, 6.05782356618264171710568495596, 6.32192236550450694976045486530, 6.68731653988993139741599168248, 7.29350762031471057748717102109, 7.81063907450705581096185251312, 7.942445275560774798604858549970

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