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Properties

Label 2-38646-1.1-c1-0-15
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s + 5·13-s + 14-s + 16-s + 3·17-s + 19-s − 3·20-s + 3·23-s + 4·25-s − 5·26-s − 28-s + 6·29-s − 7·31-s − 32-s − 3·34-s + 3·35-s + 2·37-s − 38-s + 3·40-s − 10·43-s − 3·46-s − 6·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.670·20-s + 0.625·23-s + 4/5·25-s − 0.980·26-s − 0.188·28-s + 1.11·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s + 0.507·35-s + 0.328·37-s − 0.162·38-s + 0.474·40-s − 1.52·43-s − 0.442·46-s − 6/7·49-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38646,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
19 \( 1 - T \)
113 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.31666046109783, −14.71030294664712, −14.14531424972396, −13.47655914294258, −12.81502213217563, −12.49088185382325, −11.69043234376748, −11.40334502386911, −11.03931482618479, −10.20282592135648, −9.963420379258208, −9.042972641454255, −8.655083811362924, −8.156439405064653, −7.720413770230516, −6.956363422363250, −6.713757240832012, −5.841266724717445, −5.293190745779388, −4.408271178598604, −3.689672708759210, −3.376736482571370, −2.653213738399267, −1.509251574542923, −0.8920399354326448, 0, 0.8920399354326448, 1.509251574542923, 2.653213738399267, 3.376736482571370, 3.689672708759210, 4.408271178598604, 5.293190745779388, 5.841266724717445, 6.713757240832012, 6.956363422363250, 7.720413770230516, 8.156439405064653, 8.655083811362924, 9.042972641454255, 9.963420379258208, 10.20282592135648, 11.03931482618479, 11.40334502386911, 11.69043234376748, 12.49088185382325, 12.81502213217563, 13.47655914294258, 14.14531424972396, 14.71030294664712, 15.31666046109783

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