Properties

Label 4-37e2-1.1-c1e2-0-1
Degree $4$
Conductor $1369$
Sign $1$
Analytic cond. $0.0872886$
Root an. cond. $0.543549$
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 + 2·4-s − 5-s − 2·7-s − 5·8-s + 3·9-s + 10-s − 4·11-s + 2·13-s + 2·14-s + 5·16-s − 3·17-s − 3·18-s + 6·19-s − 2·20-s + 4·22-s − 8·23-s + 5·25-s − 2·26-s − 4·28-s + 18·29-s − 20·31-s − 10·32-s + 3·34-s + 2·35-s + 6·36-s − 11·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 0.447·5-s − 0.755·7-s − 1.76·8-s + 9-s + 0.316·10-s − 1.20·11-s + 0.554·13-s + 0.534·14-s + 5/4·16-s − 0.727·17-s − 0.707·18-s + 1.37·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s + 25-s − 0.392·26-s − 0.755·28-s + 3.34·29-s − 3.59·31-s − 1.76·32-s + 0.514·34-s + 0.338·35-s + 36-s − 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1369\)    =    \(37^{2}\)
Sign: $1$
Analytic conductor: \(0.0872886\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1369,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4192326098\)
\(L(\frac12)\) \(\approx\) \(0.4192326098\)
\(L(\frac{3}{2})\) 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)$
bad37$C_2$ \( 1 + 11 T + p T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−16.30070774625329966321574399972, −15.92126559659218090507148212754, −15.89959470749858249320840417770, −15.49686103420560485601506520234, −14.45606531955039221100997664345, −13.88291242818527276459679165745, −12.86305075480989267822374846899, −12.50740797366944525930206561141, −11.98519994974156085879271117347, −11.18579217816029458093362187140, −10.32278704395388493881457457093, −10.23218905869563274534139853984, −8.997337969576225080178847698983, −8.778829562106292447356049750462, −7.51862835549577123370251104304, −7.18371500213717680724324414889, −6.27178619007762406330651043689, −5.42454872665064942861452531044, −3.81870136798848822668749225626, −2.67162594208091206392648495769, 2.67162594208091206392648495769, 3.81870136798848822668749225626, 5.42454872665064942861452531044, 6.27178619007762406330651043689, 7.18371500213717680724324414889, 7.51862835549577123370251104304, 8.778829562106292447356049750462, 8.997337969576225080178847698983, 10.23218905869563274534139853984, 10.32278704395388493881457457093, 11.18579217816029458093362187140, 11.98519994974156085879271117347, 12.50740797366944525930206561141, 12.86305075480989267822374846899, 13.88291242818527276459679165745, 14.45606531955039221100997664345, 15.49686103420560485601506520234, 15.89959470749858249320840417770, 15.92126559659218090507148212754, 16.30070774625329966321574399972

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