Properties

Label 2-38646-1.1-c1-0-2
Degree $2$
Conductor $38646$
Sign $1$
Analytic cond. $308.589$
Root an. cond. $17.5667$
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 − 4·5-s − 4·7-s + 8-s − 4·10-s − 4·14-s + 16-s + 6·17-s + 19-s − 4·20-s − 6·23-s + 11·25-s − 4·28-s + 6·29-s + 10·31-s + 32-s + 6·34-s + 16·35-s − 12·37-s + 38-s − 4·40-s − 10·41-s − 4·43-s − 6·46-s + 6·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s − 1.51·7-s + 0.353·8-s − 1.26·10-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.894·20-s − 1.25·23-s + 11/5·25-s − 0.755·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.02·34-s + 2.70·35-s − 1.97·37-s + 0.162·38-s − 0.632·40-s − 1.56·41-s − 0.609·43-s − 0.884·46-s + 0.875·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38646\)    =    \(2 \cdot 3^{2} \cdot 19 \cdot 113\)
Sign: $1$
Analytic conductor: \(308.589\)
Root analytic conductor: \(17.5667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{38646} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278565316\)
\(L(\frac12)\) \(\approx\) \(1.278565316\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
19 \( 1 - T \)
113 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.97556724975766, −14.18731322423308, −13.86247192237206, −13.24746677673062, −12.49957443156414, −12.24709843032891, −11.79652520548230, −11.59963446555315, −10.45901650788119, −10.18870072613009, −9.822776925557969, −8.658981902071964, −8.419673572663272, −7.740804818508451, −7.168686118547219, −6.701414849227935, −6.189193454190015, −5.393343827472900, −4.801983876170647, −4.018566226700316, −3.613630437081059, −3.169341818063843, −2.623763680352895, −1.296019856842938, −0.3918107643370827, 0.3918107643370827, 1.296019856842938, 2.623763680352895, 3.169341818063843, 3.613630437081059, 4.018566226700316, 4.801983876170647, 5.393343827472900, 6.189193454190015, 6.701414849227935, 7.168686118547219, 7.740804818508451, 8.419673572663272, 8.658981902071964, 9.822776925557969, 10.18870072613009, 10.45901650788119, 11.59963446555315, 11.79652520548230, 12.24709843032891, 12.49957443156414, 13.24746677673062, 13.86247192237206, 14.18731322423308, 14.97556724975766

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