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Properties

Label 4-390e2-1.1-c1e2-0-15
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 − 11-s + 5·13-s − 3·14-s + 2·15-s − 16-s + 5·19-s + 3·21-s + 22-s + 4·23-s + 24-s + 3·25-s − 5·26-s − 27-s − 2·30-s + 20·31-s − 33-s + 6·35-s + 37-s − 5·38-s + 5·39-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 − 0.301·11-s + 1.38·13-s − 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.14·19-s + 0.654·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 3/5·25-s − 0.980·26-s − 0.192·27-s − 0.365·30-s + 3.59·31-s − 0.174·33-s + 1.01·35-s + 0.164·37-s − 0.811·38-s + 0.800·39-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

\(L(1)\) \(\approx\) \(2.010491062\)
\(L(\frac12)\) \(\approx\) \(2.010491062\)
\(L(\frac{3}{2})\) 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.ad_c
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_abn
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.47.s_gt
53$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.53.ba_kp
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cz
71$C_2^2$ \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_acp
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2^2$ \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_bv
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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

−11.40655522443016623374508753724, −11.12404713016780228056349221395, −10.34764191178301719309077505078, −10.27043887772521701990008525581, −9.515933477543360842369224289040, −9.401552158342165510545022240980, −8.687890017258664319130095368439, −8.322000406059020873032228095327, −7.993971716072043059200473055327, −7.73415438426780140686718026043, −6.62330776080705882361830139222, −6.58710689688921022555872509310, −5.83154900473575940787590018557, −5.07785596282092560504841060970, −4.83044610181464682938803592506, −4.13057929401455010479128368689, −2.94655593616989399592448213886, −2.94043778047944064439256348064, −1.50504020283557303662799104360, −1.33862244464039955174558479217, 1.33862244464039955174558479217, 1.50504020283557303662799104360, 2.94043778047944064439256348064, 2.94655593616989399592448213886, 4.13057929401455010479128368689, 4.83044610181464682938803592506, 5.07785596282092560504841060970, 5.83154900473575940787590018557, 6.58710689688921022555872509310, 6.62330776080705882361830139222, 7.73415438426780140686718026043, 7.993971716072043059200473055327, 8.322000406059020873032228095327, 8.687890017258664319130095368439, 9.401552158342165510545022240980, 9.515933477543360842369224289040, 10.27043887772521701990008525581, 10.34764191178301719309077505078, 11.12404713016780228056349221395, 11.40655522443016623374508753724

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