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Properties

Label 4-311904-1.1-c1e2-0-5
Degree $4$
Conductor $311904$
Sign $1$
Analytic cond. $19.8872$
Root an. cond. $2.11175$
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  − 2-s − 3-s + 4-s + 6-s + 6·7-s − 8-s + 9-s − 12-s − 6·14-s + 16-s − 18-s − 2·19-s − 6·21-s + 24-s + 4·25-s − 27-s + 6·28-s − 6·29-s − 32-s + 36-s + 2·38-s + 8·41-s + 6·42-s − 48-s + 14·49-s − 4·50-s + 54-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 2.26·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.60·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 1.30·21-s + 0.204·24-s + 4/5·25-s − 0.192·27-s + 1.13·28-s − 1.11·29-s − 0.176·32-s + 1/6·36-s + 0.324·38-s + 1.24·41-s + 0.925·42-s − 0.144·48-s + 2·49-s − 0.565·50-s + 0.136·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311904\)    =    \(2^{5} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(19.8872\)
Root analytic conductor: \(2.11175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.399570025\)
\(L(\frac12)\) \(\approx\) \(1.399570025\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.7.ag_w
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
23$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.23.a_bg
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.g_co
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.37.a_aby
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.47.a_ae
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ag_eo
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.k_fi
67$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.67.a_adm
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.c_fm
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.79.a_afe
83$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.83.a_acc
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.89.aba_ni
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−8.833340910761533957330478798667, −8.305344627030948052465829144088, −7.81112497213833277955895124171, −7.59188784887516363482089610894, −7.12998689511588841583032193183, −6.43665312541074609380190259739, −6.00058306731053923454829630568, −5.36622138805038309129770154541, −4.98665686517280858155513796751, −4.49257862218754637012311508840, −3.97627218853148524179557715705, −3.06082397232802772881305171393, −2.10579774199186446292901728815, −1.71819293263255712388272447557, −0.842640771266003463500603695662, 0.842640771266003463500603695662, 1.71819293263255712388272447557, 2.10579774199186446292901728815, 3.06082397232802772881305171393, 3.97627218853148524179557715705, 4.49257862218754637012311508840, 4.98665686517280858155513796751, 5.36622138805038309129770154541, 6.00058306731053923454829630568, 6.43665312541074609380190259739, 7.12998689511588841583032193183, 7.59188784887516363482089610894, 7.81112497213833277955895124171, 8.305344627030948052465829144088, 8.833340910761533957330478798667

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