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Properties

Label 4-2280e2-1.1-c1e2-0-17
Degree $4$
Conductor $5198400$
Sign $1$
Analytic cond. $331.454$
Root an. cond. $4.26683$
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  + 2·3-s − 2·5-s + 6·7-s + 3·9-s + 2·11-s + 6·13-s − 4·15-s + 2·19-s + 12·21-s − 4·23-s + 3·25-s + 4·27-s + 6·29-s + 4·33-s − 12·35-s + 2·37-s + 12·39-s − 6·41-s + 6·43-s − 6·45-s − 4·47-s + 18·49-s + 8·53-s − 4·55-s + 4·57-s − 12·59-s + 18·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 2.26·7-s + 9-s + 0.603·11-s + 1.66·13-s − 1.03·15-s + 0.458·19-s + 2.61·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 0.696·33-s − 2.02·35-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.914·43-s − 0.894·45-s − 0.583·47-s + 18/7·49-s + 1.09·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s + 2.26·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5198400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(331.454\)
Root analytic conductor: \(4.26683\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5198400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.426417170\)
\(L(\frac12)\) \(\approx\) \(6.426417170\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_s
11$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_s
13$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_be
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_be
29$D_{4}$ \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_ck
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cs
41$D_{4}$ \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_bu
43$D_{4}$ \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dm
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_da
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fe
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.61.a_dy
67$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_hi
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$D_{4}$ \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_gc
97$D_{4}$ \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_je
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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

−8.813835114355619889166417595921, −8.744196797636863967857935452811, −8.423728158455591085787733275322, −8.124133534546338764572201047299, −7.71505825470713131652386875527, −7.65457525873720904458324422917, −7.03025436768784851638368602193, −6.53555781010791228863114693826, −6.21352430822398207760285743444, −5.60110665967847373077520919781, −5.02787244041845602740975144412, −4.76791339661793130876811602081, −4.16990789957791228826056429269, −4.07552928210134086867168625705, −3.36818905496675062328644755452, −3.26000788720588408447623475329, −2.15554182042046854130630956339, −2.08156270990990432656257787061, −1.15281659063469294481992806686, −1.03759875975400455200289852186, 1.03759875975400455200289852186, 1.15281659063469294481992806686, 2.08156270990990432656257787061, 2.15554182042046854130630956339, 3.26000788720588408447623475329, 3.36818905496675062328644755452, 4.07552928210134086867168625705, 4.16990789957791228826056429269, 4.76791339661793130876811602081, 5.02787244041845602740975144412, 5.60110665967847373077520919781, 6.21352430822398207760285743444, 6.53555781010791228863114693826, 7.03025436768784851638368602193, 7.65457525873720904458324422917, 7.71505825470713131652386875527, 8.124133534546338764572201047299, 8.423728158455591085787733275322, 8.744196797636863967857935452811, 8.813835114355619889166417595921

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