Properties

Label 4-36992-1.1-c1e2-0-2
Degree $4$
Conductor $36992$
Sign $1$
Analytic cond. $2.35864$
Root an. cond. $1.23926$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 2·9-s + 16-s + 6·17-s − 2·18-s − 2·25-s − 32-s − 6·34-s + 2·36-s + 14·49-s + 2·50-s + 64-s + 6·68-s − 2·72-s − 5·81-s + 12·89-s − 14·98-s − 2·100-s − 16·103-s − 14·121-s + 127-s − 128-s + 131-s − 6·136-s + 137-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1/4·16-s + 1.45·17-s − 0.471·18-s − 2/5·25-s − 0.176·32-s − 1.02·34-s + 1/3·36-s + 2·49-s + 0.282·50-s + 1/8·64-s + 0.727·68-s − 0.235·72-s − 5/9·81-s + 1.27·89-s − 1.41·98-s − 1/5·100-s − 1.57·103-s − 1.27·121-s + 0.0887·127-s − 0.0883·128-s + 0.0873·131-s − 0.514·136-s + 0.0854·137-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: $\chi_{36992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.014991940\)
\(L(\frac12)\) \(\approx\) \(1.014991940\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−10.20577814736085356952207776211, −9.884387289899397217041620201286, −9.364746929225894270075652617230, −8.835863216401317832866339248203, −8.190590441387193158826838716515, −7.75591260690702797156496136088, −7.21242822907521813396765907095, −6.79518706283797008668965856547, −5.89942135419593380940820484985, −5.57701141450097199376705821999, −4.68351373799807119232126686211, −3.90867162313150085392849000020, −3.19825512904115960807108105793, −2.20924692317133984874547287656, −1.12942124871842761561442350612, 1.12942124871842761561442350612, 2.20924692317133984874547287656, 3.19825512904115960807108105793, 3.90867162313150085392849000020, 4.68351373799807119232126686211, 5.57701141450097199376705821999, 5.89942135419593380940820484985, 6.79518706283797008668965856547, 7.21242822907521813396765907095, 7.75591260690702797156496136088, 8.190590441387193158826838716515, 8.835863216401317832866339248203, 9.364746929225894270075652617230, 9.884387289899397217041620201286, 10.20577814736085356952207776211

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