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Properties

Label 4-630e2-1.1-c1e2-0-13
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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 − 2·5-s + 5·7-s + 8-s − 3·9-s + 2·10-s + 4·13-s − 5·14-s − 16-s + 3·18-s − 2·19-s − 6·23-s + 3·25-s − 4·26-s − 6·29-s − 2·31-s − 10·35-s − 2·37-s + 2·38-s − 2·40-s + 6·41-s − 11·43-s + 6·45-s + 6·46-s − 3·47-s + 18·49-s − 3·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.894·5-s + 1.88·7-s + 0.353·8-s − 9-s + 0.632·10-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 0.707·18-s − 0.458·19-s − 1.25·23-s + 3/5·25-s − 0.784·26-s − 1.11·29-s − 0.359·31-s − 1.69·35-s − 0.328·37-s + 0.324·38-s − 0.316·40-s + 0.937·41-s − 1.67·43-s + 0.894·45-s + 0.884·46-s − 0.437·47-s + 18/7·49-s − 0.424·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9523401494\)
\(L(\frac12)\) \(\approx\) \(0.9523401494\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−10.88741426681734605243000871393, −10.52591122944593524669596684618, −10.04557647289185315862073001385, −9.183435749433055020992111056622, −9.068241217639469572189898751686, −8.517613256738800973518775491405, −8.183991809757939425599967029202, −7.87533795619137456209992011787, −7.66241324630804466876256931079, −6.99938291930451960078344501311, −6.21236589474602714734167716556, −5.98990839337433223549129974135, −5.10926627013905223942936838720, −4.96781886722876288189389393853, −4.22071208995311141910102671992, −3.75119850905001692844385292102, −3.22817420859700057861671935877, −2.06662697538065731421792078632, −1.75029111118558796715361744192, −0.61640524746042132259195005021, 0.61640524746042132259195005021, 1.75029111118558796715361744192, 2.06662697538065731421792078632, 3.22817420859700057861671935877, 3.75119850905001692844385292102, 4.22071208995311141910102671992, 4.96781886722876288189389393853, 5.10926627013905223942936838720, 5.98990839337433223549129974135, 6.21236589474602714734167716556, 6.99938291930451960078344501311, 7.66241324630804466876256931079, 7.87533795619137456209992011787, 8.183991809757939425599967029202, 8.517613256738800973518775491405, 9.068241217639469572189898751686, 9.183435749433055020992111056622, 10.04557647289185315862073001385, 10.52591122944593524669596684618, 10.88741426681734605243000871393

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