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Properties

Label 4-1305e2-1.1-c1e2-0-10
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $108.586$
Root an. cond. $3.22807$
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  + 4-s + 2·5-s + 4·7-s + 4·11-s + 4·13-s − 3·16-s + 4·19-s + 2·20-s + 4·23-s + 3·25-s + 4·28-s + 2·29-s − 4·31-s + 8·35-s + 8·37-s − 4·41-s + 8·43-s + 4·44-s − 8·47-s − 2·49-s + 4·52-s − 4·53-s + 8·55-s − 16·59-s − 4·61-s − 7·64-s + 8·65-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.20·11-s + 1.10·13-s − 3/4·16-s + 0.917·19-s + 0.447·20-s + 0.834·23-s + 3/5·25-s + 0.755·28-s + 0.371·29-s − 0.718·31-s + 1.35·35-s + 1.31·37-s − 0.624·41-s + 1.21·43-s + 0.603·44-s − 1.16·47-s − 2/7·49-s + 0.554·52-s − 0.549·53-s + 1.07·55-s − 2.08·59-s − 0.512·61-s − 7/8·64-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(108.586\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.844866734\)
\(L(\frac12)\) \(\approx\) \(4.844866734\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_cs
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_bu
67$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_dm
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_hi
79$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_ek
83$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_es
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_be
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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

−9.625900841454291880152396868980, −9.385445675641213343479916247367, −9.045744038333210674578775402860, −8.865484712951231010193715329900, −8.072025987745776821115894533478, −7.86197586207358495849286291673, −7.55837890506713340163497813183, −6.72586420824739866711297408280, −6.60996500575213862476131281066, −6.22909440419408126030761260107, −5.70256981899790712092527893901, −5.24663188247230810137767288229, −4.59712680392637223725403581272, −4.59323310609596032035838359014, −3.74441411913574919251521192013, −3.20019221173085395800896109132, −2.67225043140704148192645255234, −1.81905966326364169270266977095, −1.56534257372319950079568832720, −1.03856991848935171518904010290, 1.03856991848935171518904010290, 1.56534257372319950079568832720, 1.81905966326364169270266977095, 2.67225043140704148192645255234, 3.20019221173085395800896109132, 3.74441411913574919251521192013, 4.59323310609596032035838359014, 4.59712680392637223725403581272, 5.24663188247230810137767288229, 5.70256981899790712092527893901, 6.22909440419408126030761260107, 6.60996500575213862476131281066, 6.72586420824739866711297408280, 7.55837890506713340163497813183, 7.86197586207358495849286291673, 8.072025987745776821115894533478, 8.865484712951231010193715329900, 9.045744038333210674578775402860, 9.385445675641213343479916247367, 9.625900841454291880152396868980

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