Properties

Label 4-975e2-1.1-c1e2-0-7
Degree $4$
Conductor $950625$
Sign $1$
Analytic cond. $60.6126$
Root an. cond. $2.79023$
Motivic weight $1$
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  − 9-s − 2·11-s − 4·16-s + 4·19-s + 4·29-s − 12·31-s − 10·41-s + 5·49-s − 16·59-s + 26·61-s − 10·71-s + 6·79-s + 81-s + 30·89-s + 2·99-s + 32·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s − 0.603·11-s − 16-s + 0.917·19-s + 0.742·29-s − 2.15·31-s − 1.56·41-s + 5/7·49-s − 2.08·59-s + 3.32·61-s − 1.18·71-s + 0.675·79-s + 1/9·81-s + 3.17·89-s + 0.201·99-s + 3.06·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(950625\)    =    \(3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(60.6126\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{975} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 950625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.316805305\)
\(L(\frac12)\) \(\approx\) \(1.316805305\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−10.31735317396824768809502703642, −9.706071431802248887959096682323, −9.422242296943292539054103396031, −8.850491829005862190419234371682, −8.735931152600397430769209815601, −8.113749870469077080885296620989, −7.70514096117492222865427021471, −7.11358636170508588123127764353, −7.09344020224328043004654971087, −6.29844590597326332851426982424, −5.98898085612776500076704849768, −5.32396479233902370931894440031, −5.02313578734529331417150860930, −4.67094723650139846474220326037, −3.74830190083547619016975775396, −3.55769438444294647887380150436, −2.81640120225307199592208901711, −2.25390446665196758907232158288, −1.64623953612891172505083644100, −0.52374012234064004903235829385, 0.52374012234064004903235829385, 1.64623953612891172505083644100, 2.25390446665196758907232158288, 2.81640120225307199592208901711, 3.55769438444294647887380150436, 3.74830190083547619016975775396, 4.67094723650139846474220326037, 5.02313578734529331417150860930, 5.32396479233902370931894440031, 5.98898085612776500076704849768, 6.29844590597326332851426982424, 7.09344020224328043004654971087, 7.11358636170508588123127764353, 7.70514096117492222865427021471, 8.113749870469077080885296620989, 8.735931152600397430769209815601, 8.850491829005862190419234371682, 9.422242296943292539054103396031, 9.706071431802248887959096682323, 10.31735317396824768809502703642

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