Properties

Label 4-36992-1.1-c1e2-0-1
Degree $4$
Conductor $36992$
Sign $1$
Analytic cond. $2.35864$
Root an. cond. $1.23926$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·3-s + 4-s − 4·6-s + 8-s + 6·9-s + 12·11-s − 4·12-s + 16-s − 2·17-s + 6·18-s − 8·19-s + 12·22-s − 4·24-s − 10·25-s + 4·27-s + 32-s − 48·33-s − 2·34-s + 6·36-s − 8·38-s + 12·41-s + 16·43-s + 12·44-s − 4·48-s + 2·49-s − 10·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.63·6-s + 0.353·8-s + 2·9-s + 3.61·11-s − 1.15·12-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 1.83·19-s + 2.55·22-s − 0.816·24-s − 2·25-s + 0.769·27-s + 0.176·32-s − 8.35·33-s − 0.342·34-s + 36-s − 1.29·38-s + 1.87·41-s + 2.43·43-s + 1.80·44-s − 0.577·48-s + 2/7·49-s − 1.41·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36992\)    =    \(2^{7} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2.35864\)
Root analytic conductor: \(1.23926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9876753722\)
\(L(\frac12)\) \(\approx\) \(0.9876753722\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.84894668716717132123408330664, −9.990565731812939352470186035115, −9.286071988264369821451329617999, −9.017545129698101601080410542700, −8.241072165068691921287501783548, −7.25002689502959545343122319948, −6.65330731261113455462227374582, −6.35150301548018653136035247695, −5.99911855171913780844071238816, −5.66883563130032604301217732020, −4.61798148851481694877863510515, −4.09840146621603381404148858714, −3.90229547122613769308947697962, −2.18485014981353314837233550045, −0.984825652924800667366751863259, 0.984825652924800667366751863259, 2.18485014981353314837233550045, 3.90229547122613769308947697962, 4.09840146621603381404148858714, 4.61798148851481694877863510515, 5.66883563130032604301217732020, 5.99911855171913780844071238816, 6.35150301548018653136035247695, 6.65330731261113455462227374582, 7.25002689502959545343122319948, 8.241072165068691921287501783548, 9.017545129698101601080410542700, 9.286071988264369821451329617999, 9.990565731812939352470186035115, 10.84894668716717132123408330664

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