Properties

Label 4-350e2-1.1-c1e2-0-15
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $7.81070$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 2·7-s + 4·8-s − 4·13-s + 4·14-s + 5·16-s − 4·17-s + 8·19-s + 4·23-s − 8·26-s + 6·28-s + 4·29-s + 8·31-s + 6·32-s − 8·34-s − 4·37-s + 16·38-s − 12·41-s − 8·43-s + 8·46-s − 8·47-s + 3·49-s − 12·52-s + 12·53-s + 8·56-s + 8·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 1.10·13-s + 1.06·14-s + 5/4·16-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 1.56·26-s + 1.13·28-s + 0.742·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s − 0.657·37-s + 2.59·38-s − 1.87·41-s − 1.21·43-s + 1.17·46-s − 1.16·47-s + 3/7·49-s − 1.66·52-s + 1.64·53-s + 1.06·56-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.81070\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{350} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.137395157\)
\(L(\frac12)\) \(\approx\) \(4.137395157\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−11.70456166423393402195005342402, −11.56145200723655951154526236015, −11.05931258416115780238385745562, −10.40695359032285207903865428901, −9.903437082948784791306054620750, −9.738844466819335119483763061725, −8.752102357752807964960941028420, −8.375924916232554564968780107376, −7.88805460590994197147068727888, −7.15714501667688683184961763344, −6.86512930623403544071985121749, −6.55331813014119933292620161213, −5.47153498297892201098083367630, −5.30743194698470000684988949513, −4.77002914868893680908717302588, −4.39664725359770703549679315738, −3.48753899289058404254400516220, −2.95437356862933556349589288599, −2.28923518601867728793453841962, −1.34144829595640246257927640363, 1.34144829595640246257927640363, 2.28923518601867728793453841962, 2.95437356862933556349589288599, 3.48753899289058404254400516220, 4.39664725359770703549679315738, 4.77002914868893680908717302588, 5.30743194698470000684988949513, 5.47153498297892201098083367630, 6.55331813014119933292620161213, 6.86512930623403544071985121749, 7.15714501667688683184961763344, 7.88805460590994197147068727888, 8.375924916232554564968780107376, 8.752102357752807964960941028420, 9.738844466819335119483763061725, 9.903437082948784791306054620750, 10.40695359032285207903865428901, 11.05931258416115780238385745562, 11.56145200723655951154526236015, 11.70456166423393402195005342402

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