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Properties

Label 2-136367-1.1-c1-0-2
Degree $2$
Conductor $136367$
Sign $1$
Analytic cond. $1088.89$
Root an. cond. $32.9984$
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 + 3·8-s − 3·9-s − 4·10-s − 4·13-s − 16-s + 8·17-s + 3·18-s − 8·19-s − 4·20-s + 23-s + 11·25-s + 4·26-s − 2·29-s − 8·31-s − 5·32-s − 8·34-s + 3·36-s − 6·37-s + 8·38-s + 12·40-s + 12·41-s − 4·43-s − 12·45-s − 46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.78·5-s + 1.06·8-s − 9-s − 1.26·10-s − 1.10·13-s − 1/4·16-s + 1.94·17-s + 0.707·18-s − 1.83·19-s − 0.894·20-s + 0.208·23-s + 11/5·25-s + 0.784·26-s − 0.371·29-s − 1.43·31-s − 0.883·32-s − 1.37·34-s + 1/2·36-s − 0.986·37-s + 1.29·38-s + 1.89·40-s + 1.87·41-s − 0.609·43-s − 1.78·45-s − 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136367\)    =    \(7^{2} \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1088.89\)
Root analytic conductor: \(32.9984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 136367,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6830704110\)
\(L(\frac12)\) \(\approx\) \(0.6830704110\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.38038445836180, −12.99072457571079, −12.53018862056699, −12.24593361250293, −11.23746154498125, −10.88447411690448, −10.32877626061095, −9.985395822491958, −9.566704495859871, −9.146554722914358, −8.700558182987185, −8.270185688855002, −7.498956310804742, −7.272798868434407, −6.339325175682007, −5.890272988459880, −5.587911703334446, −4.902505800361162, −4.631537091992084, −3.601650305505954, −3.029208602563405, −2.370868958242176, −1.751279832289781, −1.351843104851457, −0.2731310036639469, 0.2731310036639469, 1.351843104851457, 1.751279832289781, 2.370868958242176, 3.029208602563405, 3.601650305505954, 4.631537091992084, 4.902505800361162, 5.587911703334446, 5.890272988459880, 6.339325175682007, 7.272798868434407, 7.498956310804742, 8.270185688855002, 8.700558182987185, 9.146554722914358, 9.566704495859871, 9.985395822491958, 10.32877626061095, 10.88447411690448, 11.23746154498125, 12.24593361250293, 12.53018862056699, 12.99072457571079, 13.38038445836180

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