Properties

Degree $4$
Conductor $4601025$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s + 2·5-s + 3·9-s + 4·11-s − 6·12-s + 4·15-s + 5·16-s − 6·20-s + 16·23-s + 3·25-s + 4·27-s − 16·31-s + 8·33-s − 9·36-s + 12·37-s − 12·44-s + 6·45-s − 16·47-s + 10·48-s − 14·49-s + 12·53-s + 8·55-s − 24·59-s − 12·60-s − 3·64-s − 8·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 0.894·5-s + 9-s + 1.20·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.34·20-s + 3.33·23-s + 3/5·25-s + 0.769·27-s − 2.87·31-s + 1.39·33-s − 3/2·36-s + 1.97·37-s − 1.80·44-s + 0.894·45-s − 2.33·47-s + 1.44·48-s − 2·49-s + 1.64·53-s + 1.07·55-s − 3.12·59-s − 1.54·60-s − 3/8·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4601025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4601025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4601025\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4601025} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4601025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.576012997\)
\(L(\frac12)\) \(\approx\) \(3.576012997\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−7.54085143340292071159442984327, −7.02617658723181735563522650723, −6.48654966834348628333399350819, −6.24012153697644363528550581310, −5.66506514104694592652509606044, −4.94734620594985484281629006959, −4.88823595164803979266926983864, −4.59542807880365420672817957846, −3.75894385007898905483554507552, −3.56799192480000330057521215275, −3.08835967698158237163354511559, −2.63990873433467548408847720531, −1.62272537170773867336279963143, −1.53827946418072491307494025034, −0.66633821317753351700694228489, 0.66633821317753351700694228489, 1.53827946418072491307494025034, 1.62272537170773867336279963143, 2.63990873433467548408847720531, 3.08835967698158237163354511559, 3.56799192480000330057521215275, 3.75894385007898905483554507552, 4.59542807880365420672817957846, 4.88823595164803979266926983864, 4.94734620594985484281629006959, 5.66506514104694592652509606044, 6.24012153697644363528550581310, 6.48654966834348628333399350819, 7.02617658723181735563522650723, 7.54085143340292071159442984327

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