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Properties

Label 4-35e4-1.1-c1e2-0-4
Degree $4$
Conductor $1500625$
Sign $1$
Analytic cond. $95.6811$
Root an. cond. $3.12756$
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 + 4-s − 4·6-s − 9-s + 4·11-s + 2·12-s − 4·13-s + 16-s + 4·17-s + 2·18-s − 8·22-s + 2·23-s + 8·26-s − 6·27-s − 2·29-s + 12·31-s + 2·32-s + 8·33-s − 8·34-s − 36-s − 8·39-s + 10·41-s − 10·43-s + 4·44-s − 4·46-s + 4·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s − 1/3·9-s + 1.20·11-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 1.70·22-s + 0.417·23-s + 1.56·26-s − 1.15·27-s − 0.371·29-s + 2.15·31-s + 0.353·32-s + 1.39·33-s − 1.37·34-s − 1/6·36-s − 1.28·39-s + 1.56·41-s − 1.52·43-s + 0.603·44-s − 0.589·46-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(95.6811\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1500625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188988593\)
\(L(\frac12)\) \(\approx\) \(1.188988593\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_s
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_w
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bt
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$D_{4}$ \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_dv
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_ef
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_ek
59$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ck
61$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_ch
67$D_{4}$ \( 1 + 22 T + 253 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.67.w_jt
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_di
73$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_fm
79$D_{4}$ \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_li
83$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_f
89$D_{4}$ \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_fz
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_hq
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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

−9.628625591722932264566027211127, −9.508420106994809935996078002804, −9.016919897597909231891323707289, −8.808610552199473868203134978649, −8.444495872034550160721314262673, −8.037906353781384409827694633443, −7.67576225175517971002514757039, −7.41456512257279403945557670395, −6.65466735339203379272403347950, −6.45648019926991244213253100313, −5.80415496924993768819501760621, −5.34011782243269297055595264148, −4.73552471897859809854297264236, −4.20784196868123022900594135744, −3.66424507225266072392019886067, −3.04346893394253018814076900055, −2.71726020941408614209350142015, −2.14317686849412090946203805086, −1.23607802281236117297335351951, −0.61920106445643496560190634896, 0.61920106445643496560190634896, 1.23607802281236117297335351951, 2.14317686849412090946203805086, 2.71726020941408614209350142015, 3.04346893394253018814076900055, 3.66424507225266072392019886067, 4.20784196868123022900594135744, 4.73552471897859809854297264236, 5.34011782243269297055595264148, 5.80415496924993768819501760621, 6.45648019926991244213253100313, 6.65466735339203379272403347950, 7.41456512257279403945557670395, 7.67576225175517971002514757039, 8.037906353781384409827694633443, 8.444495872034550160721314262673, 8.808610552199473868203134978649, 9.016919897597909231891323707289, 9.508420106994809935996078002804, 9.628625591722932264566027211127

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