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Properties

Label 4-3234e2-1.1-c1e2-0-0
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s + 2·11-s − 6·12-s − 8·13-s + 5·16-s − 6·17-s − 6·18-s − 4·22-s + 8·24-s − 3·25-s + 16·26-s − 4·27-s + 4·29-s − 8·31-s − 6·32-s − 4·33-s + 12·34-s + 9·36-s − 8·37-s + 16·39-s + 18·41-s + 8·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s − 1.73·12-s − 2.21·13-s + 5/4·16-s − 1.45·17-s − 1.41·18-s − 0.852·22-s + 1.63·24-s − 3/5·25-s + 3.13·26-s − 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.06·32-s − 0.696·33-s + 2.05·34-s + 3/2·36-s − 1.31·37-s + 2.56·39-s + 2.81·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4751537463\)
\(L(\frac12)\) \(\approx\) \(0.4751537463\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.23.a_bn
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_ck
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cw
47$D_{4}$ \( 1 - 8 T + p T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_bv
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_w
61$D_{4}$ \( 1 - 8 T + 75 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_cx
67$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_bf
71$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_gw
73$D_{4}$ \( 1 + 20 T + 218 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.73.u_ik
79$D_{4}$ \( 1 - 16 T + 215 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_ih
83$D_{4}$ \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_db
89$D_{4}$ \( 1 - 16 T + 214 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_ig
97$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_df
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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

−8.917605534230616761819348028343, −8.861829895023290411394302596832, −7.80265717588131564985471519427, −7.71964525481191952535765652756, −7.34255495612888113464009018230, −7.17297204238244352029051710425, −6.62516579549717356905779245327, −6.42693823002485659828964369960, −5.76200902751484007177084354807, −5.70882391913891838011310476851, −5.10206233197082223618877274641, −4.64998650550915022787716469638, −4.12694536639305925929978426688, −4.01831539051829615806540306675, −2.96577243358768784723279503753, −2.64347178413780953086229347103, −1.95048646589176865685364307025, −1.85772645634143644053923444708, −0.77817346615525128588605157229, −0.40411190916830400535300485387, 0.40411190916830400535300485387, 0.77817346615525128588605157229, 1.85772645634143644053923444708, 1.95048646589176865685364307025, 2.64347178413780953086229347103, 2.96577243358768784723279503753, 4.01831539051829615806540306675, 4.12694536639305925929978426688, 4.64998650550915022787716469638, 5.10206233197082223618877274641, 5.70882391913891838011310476851, 5.76200902751484007177084354807, 6.42693823002485659828964369960, 6.62516579549717356905779245327, 7.17297204238244352029051710425, 7.34255495612888113464009018230, 7.71964525481191952535765652756, 7.80265717588131564985471519427, 8.861829895023290411394302596832, 8.917605534230616761819348028343

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