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Properties

Label 4-578e2-1.1-c1e2-0-8
Degree $4$
Conductor $334084$
Sign $1$
Analytic cond. $21.3014$
Root an. cond. $2.14833$
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  + 4·3-s − 4-s + 4·5-s + 8·9-s + 4·11-s − 4·12-s − 4·13-s + 16·15-s + 16-s − 4·20-s − 8·23-s + 8·25-s + 12·27-s + 4·29-s + 16·33-s − 8·36-s − 12·37-s − 16·39-s + 8·41-s − 4·44-s + 32·45-s + 4·48-s + 4·52-s + 16·55-s − 16·60-s + 12·61-s − 64-s + ⋯
L(s)  = 1  + 2.30·3-s − 1/2·4-s + 1.78·5-s + 8/3·9-s + 1.20·11-s − 1.15·12-s − 1.10·13-s + 4.13·15-s + 1/4·16-s − 0.894·20-s − 1.66·23-s + 8/5·25-s + 2.30·27-s + 0.742·29-s + 2.78·33-s − 4/3·36-s − 1.97·37-s − 2.56·39-s + 1.24·41-s − 0.603·44-s + 4.77·45-s + 0.577·48-s + 0.554·52-s + 2.15·55-s − 2.06·60-s + 1.53·61-s − 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(334084\)    =    \(2^{2} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(21.3014\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 334084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.241362151\)
\(L(\frac12)\) \(\approx\) \(5.241362151\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_2$ \( 1 + T^{2} \)
17 \( 1 \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.3.ae_i
5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_i
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_i
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bg
29$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_i
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
37$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_cu
41$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_bg
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_cu
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_bg
73$C_2^2$ \( 1 + p^{2} T^{4} \) 2.73.a_a
79$C_2^2$ \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_lc
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.97.y_lc
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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

−10.63769325066418553207687371320, −10.10047088841092704861589224110, −9.852114423240333600247933387208, −9.484614340417278019899634804476, −9.343769346933127761131675688008, −8.618497597839144261181257048750, −8.602165267777454507626539340046, −8.087440446617658206228896015773, −7.43218602015535317984740593822, −7.06396925968699927619485114784, −6.33966332033913251521512392576, −6.12742431852151650095747160831, −5.23674830006419122946021082195, −4.92819191046774895706925954450, −3.98592927149576725026670524358, −3.79992030216038766375795112243, −2.99062112391479606820312582017, −2.36737007163493995648605793412, −2.10512276704478760303536536914, −1.34684557834792041633322747134, 1.34684557834792041633322747134, 2.10512276704478760303536536914, 2.36737007163493995648605793412, 2.99062112391479606820312582017, 3.79992030216038766375795112243, 3.98592927149576725026670524358, 4.92819191046774895706925954450, 5.23674830006419122946021082195, 6.12742431852151650095747160831, 6.33966332033913251521512392576, 7.06396925968699927619485114784, 7.43218602015535317984740593822, 8.087440446617658206228896015773, 8.602165267777454507626539340046, 8.618497597839144261181257048750, 9.343769346933127761131675688008, 9.484614340417278019899634804476, 9.852114423240333600247933387208, 10.10047088841092704861589224110, 10.63769325066418553207687371320

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