Properties

Label 4-3708-1.1-c1e2-0-0
Degree $4$
Conductor $3708$
Sign $-1$
Analytic cond. $0.236425$
Root an. cond. $0.697306$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4-s − 7·7-s + 6·9-s + 3·12-s − 4·13-s + 16-s + 19-s + 21·21-s − 2·25-s − 9·27-s + 7·28-s − 6·36-s − 3·37-s + 12·39-s + 43-s − 3·48-s + 23·49-s + 4·52-s − 3·57-s − 12·61-s − 42·63-s − 64-s + 2·67-s − 6·73-s + 6·75-s − 76-s + ⋯
L(s)  = 1  − 1.73·3-s − 1/2·4-s − 2.64·7-s + 2·9-s + 0.866·12-s − 1.10·13-s + 1/4·16-s + 0.229·19-s + 4.58·21-s − 2/5·25-s − 1.73·27-s + 1.32·28-s − 36-s − 0.493·37-s + 1.92·39-s + 0.152·43-s − 0.433·48-s + 23/7·49-s + 0.554·52-s − 0.397·57-s − 1.53·61-s − 5.29·63-s − 1/8·64-s + 0.244·67-s − 0.702·73-s + 0.692·75-s − 0.114·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3708\)    =    \(2^{2} \cdot 3^{2} \cdot 103\)
Sign: $-1$
Analytic conductor: \(0.236425\)
Root analytic conductor: \(0.697306\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3708,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
103$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 108 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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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

−12.45142127996617408126379938582, −11.90337927328674306525057081992, −11.10809491246833737433074112682, −10.33621232925751822764938822651, −9.923334500403239852609193874624, −9.598658298063641296575090737246, −8.910831833621615266926134800678, −7.57370742107072960804555749595, −6.95303205946709588301405919630, −6.36873393161333464228487463015, −5.84553951728082827322803021619, −5.07606574591929510242574115362, −4.11623608957528054560040047302, −3.07460668026695568250594068458, 0, 3.07460668026695568250594068458, 4.11623608957528054560040047302, 5.07606574591929510242574115362, 5.84553951728082827322803021619, 6.36873393161333464228487463015, 6.95303205946709588301405919630, 7.57370742107072960804555749595, 8.910831833621615266926134800678, 9.598658298063641296575090737246, 9.923334500403239852609193874624, 10.33621232925751822764938822651, 11.10809491246833737433074112682, 11.90337927328674306525057081992, 12.45142127996617408126379938582

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