Properties

Label 4-245e2-1.1-c3e2-0-8
Degree $4$
Conductor $60025$
Sign $1$
Analytic cond. $208.960$
Root an. cond. $3.80203$
Motivic weight $3$
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 + 10·3-s − 2·4-s + 10·5-s + 20·6-s + 21·9-s + 20·10-s + 66·11-s − 20·12-s + 10·13-s + 100·15-s − 20·16-s + 70·17-s + 42·18-s + 140·19-s − 20·20-s + 132·22-s − 16·23-s + 75·25-s + 20·26-s − 310·27-s − 258·29-s + 200·30-s − 20·31-s − 200·32-s + 660·33-s + 140·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.92·3-s − 1/4·4-s + 0.894·5-s + 1.36·6-s + 7/9·9-s + 0.632·10-s + 1.80·11-s − 0.481·12-s + 0.213·13-s + 1.72·15-s − 0.312·16-s + 0.998·17-s + 0.549·18-s + 1.69·19-s − 0.223·20-s + 1.27·22-s − 0.145·23-s + 3/5·25-s + 0.150·26-s − 2.20·27-s − 1.65·29-s + 1.21·30-s − 0.115·31-s − 1.10·32-s + 3.48·33-s + 0.706·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60025\)    =    \(5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(208.960\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{245} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 60025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(9.181241027\)
\(L(\frac12)\) \(\approx\) \(9.181241027\)
\(L(\frac{5}{2})\) 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)$
bad5$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
3$C_2$ \( ( 1 - 5 T + p^{3} T^{2} )^{2} \)
11$D_{4}$ \( 1 - 6 p T + 325 p T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 10 T + 19 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 70 T + 6651 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 140 T + 14218 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 16 T + 15774 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 20 T + 20082 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 328 T + 106906 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 300 T + 50342 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 30 T + 190271 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 540 T + 212254 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 380 T + 429258 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 1080 T + 705962 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 468 T + 554906 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 860 T + 522934 T^{2} + 860 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 40 T + 703974 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 240 T - 164062 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1630 T + 2133171 T^{2} - 1630 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.74587977499222614555230142234, −11.58968204306984027042945458344, −11.15330591836305629358613737650, −10.15671253723971945814386540201, −9.561655027061106917618279824170, −9.540686116183285924535022904666, −8.878326135696422137222674198250, −8.804817775980628669730559274887, −7.967570444818549368119420447570, −7.51979363449750788782525662000, −7.03911760226393773621923369053, −6.19995531418782793662909315371, −5.46802156274272187664260207126, −5.44027596967170726333397173976, −4.06453919960310431312638423250, −3.92591194441171380100827390481, −3.27028563145929960322759286153, −2.62346085043667259630895313482, −1.88561817777381399392277858905, −1.07303701354296771552814746054, 1.07303701354296771552814746054, 1.88561817777381399392277858905, 2.62346085043667259630895313482, 3.27028563145929960322759286153, 3.92591194441171380100827390481, 4.06453919960310431312638423250, 5.44027596967170726333397173976, 5.46802156274272187664260207126, 6.19995531418782793662909315371, 7.03911760226393773621923369053, 7.51979363449750788782525662000, 7.967570444818549368119420447570, 8.804817775980628669730559274887, 8.878326135696422137222674198250, 9.540686116183285924535022904666, 9.561655027061106917618279824170, 10.15671253723971945814386540201, 11.15330591836305629358613737650, 11.58968204306984027042945458344, 11.74587977499222614555230142234

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