Properties

Label 4-390e2-1.1-c1e2-0-4
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
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-s + 3-s − 2·5-s + 6-s − 3·7-s − 8-s − 2·10-s + 3·11-s − 5·13-s − 3·14-s − 2·15-s − 16-s − 3·19-s − 3·21-s + 3·22-s + 4·23-s − 24-s + 3·25-s − 5·26-s − 27-s + 4·29-s − 2·30-s + 12·31-s + 3·33-s + 6·35-s − 9·37-s − 3·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 0.632·10-s + 0.904·11-s − 1.38·13-s − 0.801·14-s − 0.516·15-s − 1/4·16-s − 0.688·19-s − 0.654·21-s + 0.639·22-s + 0.834·23-s − 0.204·24-s + 3/5·25-s − 0.980·26-s − 0.192·27-s + 0.742·29-s − 0.365·30-s + 2.15·31-s + 0.522·33-s + 1.01·35-s − 1.47·37-s − 0.486·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.749410615\)
\(L(\frac12)\) \(\approx\) \(1.749410615\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_ak
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_an
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2^2$ \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.37.j_bs
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_ch
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_cf
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_az
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_ad
71$C_2^2$ \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_ev
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.83.abg_qg
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_abh
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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

−12.12184536074245853113234315833, −11.08177851998617961590696196601, −10.68344576651312832964093446528, −10.14896411386427865856542304193, −9.646441240947636996199826738004, −9.182759533230469024555428933063, −8.942884903496133504325607963010, −8.212057093468741345461616144901, −7.948922916173447685840328622677, −7.01170411604508090564898144515, −7.00562476633892184609951392641, −6.35588237001831651787079776003, −5.84610574046295629904311059978, −4.90849425690920176090543987196, −4.69164564802427884901286656054, −3.76996627205037880294805430924, −3.75353878660199665088185467812, −2.61475015982717588378116578525, −2.60647991919117434963431003928, −0.76962871265009272834052813845, 0.76962871265009272834052813845, 2.60647991919117434963431003928, 2.61475015982717588378116578525, 3.75353878660199665088185467812, 3.76996627205037880294805430924, 4.69164564802427884901286656054, 4.90849425690920176090543987196, 5.84610574046295629904311059978, 6.35588237001831651787079776003, 7.00562476633892184609951392641, 7.01170411604508090564898144515, 7.948922916173447685840328622677, 8.212057093468741345461616144901, 8.942884903496133504325607963010, 9.182759533230469024555428933063, 9.646441240947636996199826738004, 10.14896411386427865856542304193, 10.68344576651312832964093446528, 11.08177851998617961590696196601, 12.12184536074245853113234315833

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