Properties

Label 4-1805e2-1.1-c1e2-0-0
Degree $4$
Conductor $3258025$
Sign $1$
Analytic cond. $207.734$
Root an. cond. $3.79644$
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  − 2·5-s + 6·9-s − 2·11-s − 4·16-s − 25-s − 18·29-s − 14·31-s + 4·41-s − 12·45-s − 2·49-s + 4·55-s − 18·59-s − 14·61-s + 2·71-s − 2·79-s + 8·80-s + 27·81-s + 22·89-s − 12·99-s + 30·101-s − 30·109-s − 19·121-s + 12·125-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s − 0.603·11-s − 16-s − 1/5·25-s − 3.34·29-s − 2.51·31-s + 0.624·41-s − 1.78·45-s − 2/7·49-s + 0.539·55-s − 2.34·59-s − 1.79·61-s + 0.237·71-s − 0.225·79-s + 0.894·80-s + 3·81-s + 2.33·89-s − 1.20·99-s + 2.98·101-s − 2.87·109-s − 1.72·121-s + 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3258025\)    =    \(5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(207.734\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3258025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6607445400\)
\(L(\frac12)\) \(\approx\) \(0.6607445400\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_2$ \( 1 + 2 T + p T^{2} \)
19 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.59.s_hr
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.71.ac_fn
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.89.aw_ln
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−9.517336983817381026615188574883, −9.043107785801728578803226425845, −9.017668135986333554840430619668, −7.942953648120923239406336837282, −7.83152404885177440445066360317, −7.48716003058162486652418858493, −7.20610153750721872675018328112, −7.00590706481127108489076837400, −6.09830437590725519054301895606, −6.07954418908341089131109404540, −5.26060202433194636492583668451, −4.94578460729255212020682293118, −4.48592941136036888675538488598, −3.97707571169612693924667495769, −3.71066866067635643536738663027, −3.36108865552823585813803934330, −2.42031440845551366904758511332, −1.72502237699623100400773022357, −1.66364211856435982528954261813, −0.29250181756487365504488363043, 0.29250181756487365504488363043, 1.66364211856435982528954261813, 1.72502237699623100400773022357, 2.42031440845551366904758511332, 3.36108865552823585813803934330, 3.71066866067635643536738663027, 3.97707571169612693924667495769, 4.48592941136036888675538488598, 4.94578460729255212020682293118, 5.26060202433194636492583668451, 6.07954418908341089131109404540, 6.09830437590725519054301895606, 7.00590706481127108489076837400, 7.20610153750721872675018328112, 7.48716003058162486652418858493, 7.83152404885177440445066360317, 7.942953648120923239406336837282, 9.017668135986333554840430619668, 9.043107785801728578803226425845, 9.517336983817381026615188574883

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