Properties

Label 4-177e2-1.1-c3e2-0-0
Degree $4$
Conductor $31329$
Sign $1$
Analytic cond. $109.063$
Root an. cond. $3.23161$
Motivic weight $3$
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  + 7·3-s − 16·4-s + 58·7-s + 22·9-s − 112·12-s + 192·16-s + 238·19-s + 406·21-s + 191·25-s − 35·27-s − 928·28-s − 352·36-s + 1.34e3·48-s + 1.83e3·49-s + 1.66e3·57-s + 1.27e3·63-s − 2.04e3·64-s + 1.33e3·75-s − 3.80e3·76-s − 1.67e3·79-s − 839·81-s − 6.49e3·84-s − 3.05e3·100-s + 560·108-s + 1.11e4·112-s − 2.66e3·121-s + 127-s + ⋯
L(s)  = 1  + 1.34·3-s − 2·4-s + 3.13·7-s + 0.814·9-s − 2.69·12-s + 3·16-s + 2.87·19-s + 4.21·21-s + 1.52·25-s − 0.249·27-s − 6.26·28-s − 1.62·36-s + 4.04·48-s + 5.35·49-s + 3.87·57-s + 2.55·63-s − 4·64-s + 2.05·75-s − 5.74·76-s − 2.37·79-s − 1.15·81-s − 8.43·84-s − 3.05·100-s + 0.498·108-s + 9.39·112-s − 2·121-s + 0.000698·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31329\)    =    \(3^{2} \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(109.063\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 31329,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.084451204\)
\(L(\frac12)\) \(\approx\) \(4.084451204\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - 7 T + p^{3} T^{2} \)
59$C_2$ \( 1 + p^{3} T^{2} \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
5$C_2$ \( ( 1 - 21 T + p^{3} T^{2} )( 1 + 21 T + p^{3} T^{2} ) \)
7$C_2$ \( ( 1 - 29 T + p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 126 T + p^{3} T^{2} )( 1 + 126 T + p^{3} T^{2} ) \)
19$C_2$ \( ( 1 - 119 T + p^{3} T^{2} )^{2} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 159 T + p^{3} T^{2} )( 1 + 159 T + p^{3} T^{2} ) \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 525 T + p^{3} T^{2} )( 1 + 525 T + p^{3} T^{2} ) \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 327 T + p^{3} T^{2} )( 1 + 327 T + p^{3} T^{2} ) \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 180 T + p^{3} T^{2} )( 1 + 180 T + p^{3} T^{2} ) \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 835 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
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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

−12.49338607896575808882919387680, −12.09754900734853022406129674921, −11.32714976573786649359600167782, −11.22112249526144457312122795401, −10.20781784494847144436184958513, −9.928500073106134652354494539438, −9.098158396872551140033711569384, −8.918114637909309176958437661475, −8.480523625788813089939296677648, −7.952062087992253160956052167368, −7.70221821928741192311864170520, −7.27134637775394327836645570978, −5.59738740432383147152077682183, −5.22893986232767485877879661595, −4.82408733762506964925126528470, −4.31829873925310447425772956738, −3.57348280875314381547334040845, −2.79274501633716833900179369085, −1.44273421242412220127641180227, −1.09885709454615079615261044027, 1.09885709454615079615261044027, 1.44273421242412220127641180227, 2.79274501633716833900179369085, 3.57348280875314381547334040845, 4.31829873925310447425772956738, 4.82408733762506964925126528470, 5.22893986232767485877879661595, 5.59738740432383147152077682183, 7.27134637775394327836645570978, 7.70221821928741192311864170520, 7.952062087992253160956052167368, 8.480523625788813089939296677648, 8.918114637909309176958437661475, 9.098158396872551140033711569384, 9.928500073106134652354494539438, 10.20781784494847144436184958513, 11.22112249526144457312122795401, 11.32714976573786649359600167782, 12.09754900734853022406129674921, 12.49338607896575808882919387680

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