Properties

Label 4-1110e2-1.1-c1e2-0-4
Degree $4$
Conductor $1232100$
Sign $1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
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  + 2·2-s + 3·4-s − 4·5-s + 4·8-s − 9-s − 8·10-s − 6·11-s + 8·13-s + 5·16-s − 14·17-s − 2·18-s − 12·20-s − 12·22-s − 12·23-s + 11·25-s + 16·26-s + 6·32-s − 28·34-s − 3·36-s − 12·37-s − 16·40-s + 14·41-s − 2·43-s − 18·44-s + 4·45-s − 24·46-s + 5·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.41·8-s − 1/3·9-s − 2.52·10-s − 1.80·11-s + 2.21·13-s + 5/4·16-s − 3.39·17-s − 0.471·18-s − 2.68·20-s − 2.55·22-s − 2.50·23-s + 11/5·25-s + 3.13·26-s + 1.06·32-s − 4.80·34-s − 1/2·36-s − 1.97·37-s − 2.52·40-s + 2.18·41-s − 0.304·43-s − 2.71·44-s + 0.596·45-s − 3.53·46-s + 5/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614357300\)
\(L(\frac12)\) \(\approx\) \(1.614357300\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_2$ \( 1 + 12 T + p T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−10.50481981018230467959420619220, −9.734147090374738336236109399737, −8.928070521712944586832036579470, −8.610783437927459451696537297198, −8.366668653066126170496621987381, −7.959606861417185467106879037706, −7.55055457524199038899952707646, −7.06911250900582908531332821944, −6.58280040404106115335816011621, −6.12964104100658846555451893656, −5.95474126877153250044088212120, −5.00294905636482153606728987648, −4.97212452293138690184969758999, −4.13407548213147831280632398367, −3.84664529118434080072232867263, −3.83205948541746557387053060939, −2.89766463121113304753673570649, −2.38792154522695698696898176337, −1.89667312577178878410517798237, −0.41617444926428710550282091252, 0.41617444926428710550282091252, 1.89667312577178878410517798237, 2.38792154522695698696898176337, 2.89766463121113304753673570649, 3.83205948541746557387053060939, 3.84664529118434080072232867263, 4.13407548213147831280632398367, 4.97212452293138690184969758999, 5.00294905636482153606728987648, 5.95474126877153250044088212120, 6.12964104100658846555451893656, 6.58280040404106115335816011621, 7.06911250900582908531332821944, 7.55055457524199038899952707646, 7.959606861417185467106879037706, 8.366668653066126170496621987381, 8.610783437927459451696537297198, 8.928070521712944586832036579470, 9.734147090374738336236109399737, 10.50481981018230467959420619220

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