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Properties

Label 4-1805e2-1.1-c1e2-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 2·5-s + 6·9-s − 8·11-s + 5·16-s − 6·20-s − 25-s − 12·29-s + 8·31-s + 18·36-s + 20·41-s − 24·44-s − 12·45-s + 10·49-s + 16·55-s + 4·61-s + 3·64-s − 8·71-s + 8·79-s − 10·80-s + 27·81-s − 4·89-s − 48·99-s − 3·100-s − 12·101-s + 12·109-s − 36·116-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s + 2·9-s − 2.41·11-s + 5/4·16-s − 1.34·20-s − 1/5·25-s − 2.22·29-s + 1.43·31-s + 3·36-s + 3.12·41-s − 3.61·44-s − 1.78·45-s + 10/7·49-s + 2.15·55-s + 0.512·61-s + 3/8·64-s − 0.949·71-s + 0.900·79-s − 1.11·80-s + 3·81-s − 0.423·89-s − 4.82·99-s − 0.299·100-s − 1.19·101-s + 1.14·109-s − 3.34·116-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

\(L(1)\) \(\approx\) \(2.703280711\)
\(L(\frac12)\) \(\approx\) \(2.703280711\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
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$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \) 2.83.a_gc
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
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

−9.363847935394736232068273008489, −9.362531116971785895102511547506, −8.468562224646369129477604645478, −7.950771934631117162400433233461, −7.80498990938258953444581486176, −7.44132017131287917434802625896, −7.25056949826501419901367460831, −6.98760151886358534286511560451, −6.23925995807841324328217601146, −5.93921632520865009025170395212, −5.50611793637179759475933898353, −4.94231025974138457774315043763, −4.52349203961254367940021236201, −3.90598276121166886821548053738, −3.78153311591346673780064539915, −2.84280678188279091484827638574, −2.53916964481262581412084284365, −2.13120344577616346474351662927, −1.43732905366755316359508533950, −0.60121922839341753337741176985, 0.60121922839341753337741176985, 1.43732905366755316359508533950, 2.13120344577616346474351662927, 2.53916964481262581412084284365, 2.84280678188279091484827638574, 3.78153311591346673780064539915, 3.90598276121166886821548053738, 4.52349203961254367940021236201, 4.94231025974138457774315043763, 5.50611793637179759475933898353, 5.93921632520865009025170395212, 6.23925995807841324328217601146, 6.98760151886358534286511560451, 7.25056949826501419901367460831, 7.44132017131287917434802625896, 7.80498990938258953444581486176, 7.950771934631117162400433233461, 8.468562224646369129477604645478, 9.362531116971785895102511547506, 9.363847935394736232068273008489

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