↑

Properties

Label 4-6046e2-1.1-c1e2-0-0
Degree $4$
Conductor $36554116$
Sign $1$
Analytic cond. $2330.72$
Root an. cond. $6.94820$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 6·5-s − 6·6-s − 5·7-s + 4·8-s + 2·9-s − 12·10-s − 3·11-s − 9·12-s − 8·13-s − 10·14-s + 18·15-s + 5·16-s − 10·17-s + 4·18-s − 11·19-s − 18·20-s + 15·21-s − 6·22-s − 8·23-s − 12·24-s + 17·25-s − 16·26-s + 6·27-s − 15·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.68·5-s − 2.44·6-s − 1.88·7-s + 1.41·8-s + 2/3·9-s − 3.79·10-s − 0.904·11-s − 2.59·12-s − 2.21·13-s − 2.67·14-s + 4.64·15-s + 5/4·16-s − 2.42·17-s + 0.942·18-s − 2.52·19-s − 4.02·20-s + 3.27·21-s − 1.27·22-s − 1.66·23-s − 2.44·24-s + 17/5·25-s − 3.13·26-s + 1.15·27-s − 2.83·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36554116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36554116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36554116\)    =    \(2^{2} \cdot 3023^{2}\)
Sign: $1$
Analytic conductor: \(2330.72\)
Root analytic conductor: \(6.94820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((4,\ 36554116,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3023 \( 1+O(T) \)
good3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 13 T + 93 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 105 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 89 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 103 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 178 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.14986406050398670056444818843, −7.10116666872104284257900660537, −6.61815696621785750282684977632, −6.54582661909121088522910605655, −6.05030129321244386549756197301, −5.97644727298919479394976097139, −5.06646470461001858014284908912, −4.98666365586675883734657485477, −4.66820644242868404056139431046, −4.32846728345785024807493335882, −3.87670975025941844630570564584, −3.80269956502424158457407770484, −3.03223572917756441375214731217, −2.80207683302368755476818249028, −2.35127339045934623827273323973, −1.84780794084992231694967041801, 0, 0, 0, 0, 1.84780794084992231694967041801, 2.35127339045934623827273323973, 2.80207683302368755476818249028, 3.03223572917756441375214731217, 3.80269956502424158457407770484, 3.87670975025941844630570564584, 4.32846728345785024807493335882, 4.66820644242868404056139431046, 4.98666365586675883734657485477, 5.06646470461001858014284908912, 5.97644727298919479394976097139, 6.05030129321244386549756197301, 6.54582661909121088522910605655, 6.61815696621785750282684977632, 7.10116666872104284257900660537, 7.14986406050398670056444818843

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