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Properties

Label 4-1190e2-1.1-c1e2-0-5
Degree $4$
Conductor $1416100$
Sign $1$
Analytic cond. $90.2917$
Root an. cond. $3.08256$
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  − 2·2-s + 3-s + 3·4-s − 2·5-s − 2·6-s − 2·7-s − 4·8-s − 4·9-s + 4·10-s − 3·11-s + 3·12-s + 4·13-s + 4·14-s − 2·15-s + 5·16-s + 2·17-s + 8·18-s + 19-s − 6·20-s − 2·21-s + 6·22-s + 3·23-s − 4·24-s + 3·25-s − 8·26-s − 6·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.26·10-s − 0.904·11-s + 0.866·12-s + 1.10·13-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 0.485·17-s + 1.88·18-s + 0.229·19-s − 1.34·20-s − 0.436·21-s + 1.27·22-s + 0.625·23-s − 0.816·24-s + 3/5·25-s − 1.56·26-s − 1.15·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1416100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1416100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1416100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(90.2917\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1416100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8658043181\)
\(L(\frac12)\) \(\approx\) \(0.8658043181\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
11$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_n
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_k
19$D_{4}$ \( 1 - T + 37 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_bl
23$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_r
29$D_{4}$ \( 1 - 3 T + 49 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_bx
31$D_{4}$ \( 1 - 9 T + 81 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.31.aj_dd
37$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.37.ao_eo
41$D_{4}$ \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_cb
43$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_bl
47$D_{4}$ \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.47.l_dp
53$D_{4}$ \( 1 - 17 T + 167 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.53.ar_gl
59$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.59.af_cl
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.61.q_gk
67$D_{4}$ \( 1 - 11 T + 103 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.67.al_dz
71$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_du
73$D_{4}$ \( 1 - 24 T + 270 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.73.ay_kk
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_ek
97$D_{4}$ \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.97.ay_mg
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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

−9.895263412647910930933652982072, −9.414509694112822754296633537643, −8.961160528431466087238150298261, −8.865874634041693557623422445286, −8.176204801375672867653991187295, −8.134592825295155417069053075167, −7.70349891649080801444665409769, −7.43530710026848675193628801367, −6.57946697583959377468704365654, −6.45422358936658178761733490084, −5.92073612316628727839595628025, −5.51863956820141655180037453940, −4.84887934040511287936898985568, −4.17889696894589455750568958264, −3.45271531612820729040525271488, −3.21759944033804243124736439045, −2.53278891468990519537232574473, −2.46387945864077855749066304371, −1.03127213418727248702140444369, −0.60281490246450573427858039308, 0.60281490246450573427858039308, 1.03127213418727248702140444369, 2.46387945864077855749066304371, 2.53278891468990519537232574473, 3.21759944033804243124736439045, 3.45271531612820729040525271488, 4.17889696894589455750568958264, 4.84887934040511287936898985568, 5.51863956820141655180037453940, 5.92073612316628727839595628025, 6.45422358936658178761733490084, 6.57946697583959377468704365654, 7.43530710026848675193628801367, 7.70349891649080801444665409769, 8.134592825295155417069053075167, 8.176204801375672867653991187295, 8.865874634041693557623422445286, 8.961160528431466087238150298261, 9.414509694112822754296633537643, 9.895263412647910930933652982072

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