Properties

Label 4-414e2-1.1-c1e2-0-0
Degree $4$
Conductor $171396$
Sign $1$
Analytic cond. $10.9283$
Root an. cond. $1.81818$
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-s + 3·3-s − 4·5-s + 3·6-s − 8-s + 6·9-s − 4·10-s + 3·11-s − 4·13-s − 12·15-s − 16-s + 14·17-s + 6·18-s − 10·19-s + 3·22-s − 23-s − 3·24-s + 5·25-s − 4·26-s + 9·27-s + 4·29-s − 12·30-s + 2·31-s + 9·33-s + 14·34-s + 16·37-s − 10·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1.78·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.26·10-s + 0.904·11-s − 1.10·13-s − 3.09·15-s − 1/4·16-s + 3.39·17-s + 1.41·18-s − 2.29·19-s + 0.639·22-s − 0.208·23-s − 0.612·24-s + 25-s − 0.784·26-s + 1.73·27-s + 0.742·29-s − 2.19·30-s + 0.359·31-s + 1.56·33-s + 2.40·34-s + 2.63·37-s − 1.62·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(171396\)    =    \(2^{2} \cdot 3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(10.9283\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 171396,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.202757297\)
\(L(\frac12)\) \(\approx\) \(3.202757297\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
23$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−11.89694088610312784931724861751, −11.00887440719676466154542199792, −10.47754533640344624220151454168, −9.961474013929332202881923132184, −9.579008523474239910321945895614, −9.195275006362543819238279413250, −8.402506898665240926891753789906, −8.223085702316691053359629685518, −7.85062383936103664169981478351, −7.42319122245833173013012470687, −7.09549505499549925220111423174, −6.18509693702628057546263591449, −5.78018470029439707654948468699, −4.73411107136785871359875137962, −4.30408529317748029917261864099, −3.99111344998293195475884481248, −3.50853392532970637887914749328, −2.92728258201818817031841467057, −2.34434885812628777657352552928, −1.03182151745784563105900605671, 1.03182151745784563105900605671, 2.34434885812628777657352552928, 2.92728258201818817031841467057, 3.50853392532970637887914749328, 3.99111344998293195475884481248, 4.30408529317748029917261864099, 4.73411107136785871359875137962, 5.78018470029439707654948468699, 6.18509693702628057546263591449, 7.09549505499549925220111423174, 7.42319122245833173013012470687, 7.85062383936103664169981478351, 8.223085702316691053359629685518, 8.402506898665240926891753789906, 9.195275006362543819238279413250, 9.579008523474239910321945895614, 9.961474013929332202881923132184, 10.47754533640344624220151454168, 11.00887440719676466154542199792, 11.89694088610312784931724861751

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