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Properties

Label 4-93e2-1.1-c1e2-0-3
Degree $4$
Conductor $8649$
Sign $1$
Analytic cond. $0.551467$
Root an. cond. $0.861747$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s − 4·5-s + 6·6-s − 4·7-s − 3·8-s + 3·9-s + 12·10-s − 6·11-s − 8·12-s − 2·13-s + 12·14-s + 8·15-s + 3·16-s − 4·17-s − 9·18-s − 8·19-s − 16·20-s + 8·21-s + 18·22-s + 2·23-s + 6·24-s + 7·25-s + 6·26-s − 4·27-s − 16·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s − 1.06·8-s + 9-s + 3.79·10-s − 1.80·11-s − 2.30·12-s − 0.554·13-s + 3.20·14-s + 2.06·15-s + 3/4·16-s − 0.970·17-s − 2.12·18-s − 1.83·19-s − 3.57·20-s + 1.74·21-s + 3.83·22-s + 0.417·23-s + 1.22·24-s + 7/5·25-s + 1.17·26-s − 0.769·27-s − 3.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8649\)    =    \(3^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(0.551467\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8649,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
5$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_j
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_ba
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_s
19$D_{4}$ \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bx
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bq
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_cc
37$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_be
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.41.a_bl
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_by
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_da
53$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.53.a_ba
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.71.as_ip
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_fm
79$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_fy
83$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_le
89$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_dy
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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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

−13.28325049766013466577581942696, −13.11280807102653637204140655201, −12.40139029115471144093406936955, −12.20958483103405601939915029202, −11.31027016631457569448627327080, −10.86571364150307016472537442948, −10.51499814339291424302627438531, −10.03053886868186514601515193887, −9.446206329583624367338010042651, −8.789127965445751695602008842038, −8.097901540609051783360091553900, −7.908471012343790082513713795375, −6.96212014036167050528794276049, −6.76607666390121230126886333033, −5.76260028748807662172499793565, −4.76830701724373158873311740635, −3.93958617827393882859272252084, −2.71523764769336699533400004892, 0, 0, 2.71523764769336699533400004892, 3.93958617827393882859272252084, 4.76830701724373158873311740635, 5.76260028748807662172499793565, 6.76607666390121230126886333033, 6.96212014036167050528794276049, 7.908471012343790082513713795375, 8.097901540609051783360091553900, 8.789127965445751695602008842038, 9.446206329583624367338010042651, 10.03053886868186514601515193887, 10.51499814339291424302627438531, 10.86571364150307016472537442948, 11.31027016631457569448627327080, 12.20958483103405601939915029202, 12.40139029115471144093406936955, 13.11280807102653637204140655201, 13.28325049766013466577581942696

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