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Properties

Label 4-65e4-1.1-c1e2-0-4
Degree $4$
Conductor $17850625$
Sign $1$
Analytic cond. $1138.17$
Root an. cond. $5.80833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 2·6-s − 2·7-s − 8-s − 3·9-s + 4·11-s + 2·14-s − 16-s − 8·17-s + 3·18-s + 4·19-s + 4·21-s − 4·22-s + 6·23-s + 2·24-s + 14·27-s + 2·29-s + 8·31-s + 6·32-s − 8·33-s + 8·34-s − 4·38-s − 6·41-s − 4·42-s + 6·43-s − 6·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 0.816·6-s − 0.755·7-s − 0.353·8-s − 9-s + 1.20·11-s + 0.534·14-s − 1/4·16-s − 1.94·17-s + 0.707·18-s + 0.917·19-s + 0.872·21-s − 0.852·22-s + 1.25·23-s + 0.408·24-s + 2.69·27-s + 0.371·29-s + 1.43·31-s + 1.06·32-s − 1.39·33-s + 1.37·34-s − 0.648·38-s − 0.937·41-s − 0.617·42-s + 0.914·43-s − 0.884·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17850625\)    =    \(5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1138.17\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17850625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_b
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_n
17$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bl
19$D_{4}$ \( 1 - 4 T + 29 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bd
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_h
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.37.a_cj
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_br
47$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_bu
53$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_cs
59$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.59.a_b
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$D_{4}$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_cj
73$D_{4}$ \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_gc
79$D_{4}$ \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_eg
83$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_eo
89$D_{4}$ \( 1 + 2 T + 127 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_ex
97$D_{4}$ \( 1 + 24 T + 325 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.97.y_mn
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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

−8.201658100197375726563428309566, −8.138974301942529904026999769557, −7.39452526759548797405133795775, −6.81812031924390525880533038960, −6.69753802299939821430336674733, −6.52333437943031019473710000432, −6.05317710539572665001307904888, −5.84405496272501099179071671240, −5.28840309853066241515613712858, −4.88328377783995194228477298854, −4.45386249328096668899820795977, −4.29404482096101849856557035076, −3.43260913291972924395840909618, −2.94158310033743530616353188925, −2.88936619363841194682634407621, −2.25334640849633247991995982842, −1.33419092098292340565396101352, −0.936578016626246354578999037112, 0, 0, 0.936578016626246354578999037112, 1.33419092098292340565396101352, 2.25334640849633247991995982842, 2.88936619363841194682634407621, 2.94158310033743530616353188925, 3.43260913291972924395840909618, 4.29404482096101849856557035076, 4.45386249328096668899820795977, 4.88328377783995194228477298854, 5.28840309853066241515613712858, 5.84405496272501099179071671240, 6.05317710539572665001307904888, 6.52333437943031019473710000432, 6.69753802299939821430336674733, 6.81812031924390525880533038960, 7.39452526759548797405133795775, 8.138974301942529904026999769557, 8.201658100197375726563428309566

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