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Properties

Label 4-2175e2-1.1-c1e2-0-7
Degree $4$
Conductor $4730625$
Sign $1$
Analytic cond. $301.628$
Root an. cond. $4.16742$
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-s − 2·3-s − 2·4-s − 2·6-s + 6·7-s − 3·8-s + 3·9-s − 2·11-s + 4·12-s + 8·13-s + 6·14-s + 16-s + 2·17-s + 3·18-s + 2·19-s − 12·21-s − 2·22-s + 6·24-s + 8·26-s − 4·27-s − 12·28-s + 2·29-s − 16·31-s + 2·32-s + 4·33-s + 2·34-s − 6·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s + 2.26·7-s − 1.06·8-s + 9-s − 0.603·11-s + 1.15·12-s + 2.21·13-s + 1.60·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.458·19-s − 2.61·21-s − 0.426·22-s + 1.22·24-s + 1.56·26-s − 0.769·27-s − 2.26·28-s + 0.371·29-s − 2.87·31-s + 0.353·32-s + 0.696·33-s + 0.342·34-s − 36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4730625\)    =    \(3^{2} \cdot 5^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(301.628\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4730625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.954022616\)
\(L(\frac12)\) \(\approx\) \(2.954022616\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_d
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bl
17$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_p
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ag
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_de
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \) 2.47.a_bx
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.53.as_ha
59$D_{4}$ \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_es
61$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_da
67$D_{4}$ \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_gh
71$D_{4}$ \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_gg
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_cg
83$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_fa
89$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \) 2.89.a_cb
97$D_{4}$ \( 1 + 10 T + 214 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_ig
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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

−9.026588169030694477499478980362, −9.005325148257090495466392331342, −8.290919692330478242497736919455, −8.088967505748253575289917696980, −7.934929449936648190701692076849, −7.32782091847140597912658731035, −6.84716087171808657233360422866, −6.38509201301356461873748604601, −5.73880067927267955901603094071, −5.53654177963471210352652827878, −5.25740079421440975719107165275, −5.09331683070262819187790497779, −4.37976812030464659588093840041, −4.15294009994955531876244781101, −3.76289720310333795649125104752, −3.37004892093017059497600659877, −2.29670366348512898094645852693, −1.82020158348520046433299669044, −1.04825082876271284007165115233, −0.75887526150236127825243908358, 0.75887526150236127825243908358, 1.04825082876271284007165115233, 1.82020158348520046433299669044, 2.29670366348512898094645852693, 3.37004892093017059497600659877, 3.76289720310333795649125104752, 4.15294009994955531876244781101, 4.37976812030464659588093840041, 5.09331683070262819187790497779, 5.25740079421440975719107165275, 5.53654177963471210352652827878, 5.73880067927267955901603094071, 6.38509201301356461873748604601, 6.84716087171808657233360422866, 7.32782091847140597912658731035, 7.934929449936648190701692076849, 8.088967505748253575289917696980, 8.290919692330478242497736919455, 9.005325148257090495466392331342, 9.026588169030694477499478980362

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