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Properties

Label 4-464648-1.1-c1e2-0-0
Degree $4$
Conductor $464648$
Sign $1$
Analytic cond. $29.6263$
Root an. cond. $2.33302$
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 + 4·3-s + 4-s + 4·6-s + 8-s + 6·9-s + 4·12-s + 16-s + 6·18-s + 4·24-s − 25-s − 4·27-s + 12·29-s + 32-s + 6·36-s + 18·41-s + 6·47-s + 4·48-s − 49-s − 50-s − 4·54-s + 12·58-s − 12·59-s + 8·61-s + 64-s + 2·67-s + 6·72-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.353·8-s + 2·9-s + 1.15·12-s + 1/4·16-s + 1.41·18-s + 0.816·24-s − 1/5·25-s − 0.769·27-s + 2.22·29-s + 0.176·32-s + 36-s + 2.81·41-s + 0.875·47-s + 0.577·48-s − 1/7·49-s − 0.141·50-s − 0.544·54-s + 1.57·58-s − 1.56·59-s + 1.02·61-s + 1/8·64-s + 0.244·67-s + 0.707·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(464648\)    =    \(2^{3} \cdot 241^{2}\)
Sign: $1$
Analytic conductor: \(29.6263\)
Root analytic conductor: \(2.33302\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 464648,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.767012065\)
\(L(\frac12)\) \(\approx\) \(6.767012065\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
241$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.19.a_af
23$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.23.a_ah
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.29.am_dh
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.37.a_w
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.ag_w
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ai_ez
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.ac_adm
71$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.71.a_acd
73$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.73.a_ao
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.ao_hq
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ag_gk
89$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.89.a_cw
97$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.w_kt
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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

−8.548113480210614022368436402664, −8.015340489693476527030671624522, −7.80980092161500433265759728024, −7.46311743902618040611535758833, −6.69725166545105596985166667052, −6.36749988950318584506075574390, −5.69691822193763031623385487664, −5.24756841906645773194987052893, −4.37649366294576116971858953700, −4.15816476330247363110493063544, −3.52069852969482957770985598719, −2.99173049970091757881156332247, −2.53192047695215617541765365166, −2.26539493113472070506154574566, −1.19186899166079723193862105222, 1.19186899166079723193862105222, 2.26539493113472070506154574566, 2.53192047695215617541765365166, 2.99173049970091757881156332247, 3.52069852969482957770985598719, 4.15816476330247363110493063544, 4.37649366294576116971858953700, 5.24756841906645773194987052893, 5.69691822193763031623385487664, 6.36749988950318584506075574390, 6.69725166545105596985166667052, 7.46311743902618040611535758833, 7.80980092161500433265759728024, 8.015340489693476527030671624522, 8.548113480210614022368436402664

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