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Properties

Label 2-164-164.127-c0-0-0
Degree $2$
Conductor $164$
Sign $0.997 + 0.0667i$
Analytic cond. $0.0818466$
Root an. cond. $0.286088$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.5 − 1.53i)5-s + (0.309 + 0.951i)8-s − 9-s + (0.5 + 1.53i)10-s + (1.11 + 1.53i)13-s + (−0.809 − 0.587i)16-s + (−1.11 + 0.363i)17-s + (0.809 − 0.587i)18-s + (−1.30 − 0.951i)20-s + (−1.30 − 0.951i)25-s + (−1.80 − 0.587i)26-s + 32-s + (0.690 − 0.951i)34-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.5 − 1.53i)5-s + (0.309 + 0.951i)8-s − 9-s + (0.5 + 1.53i)10-s + (1.11 + 1.53i)13-s + (−0.809 − 0.587i)16-s + (−1.11 + 0.363i)17-s + (0.809 − 0.587i)18-s + (−1.30 − 0.951i)20-s + (−1.30 − 0.951i)25-s + (−1.80 − 0.587i)26-s + 32-s + (0.690 − 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.997 + 0.0667i$
Analytic conductor: \(0.0818466\)
Root analytic conductor: \(0.286088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :0),\ 0.997 + 0.0667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4836251595\)
\(L(\frac12)\) \(\approx\) \(0.4836251595\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + 0.618T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
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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

−13.36454637750590006089461080590, −11.88032912583393608308172291929, −11.05081372759508005244730798043, −9.632719146874707793402677145655, −8.711166280378404405933828040312, −8.488002706487949380527181328802, −6.65646052297742784264411778986, −5.71967667474643560245412925019, −4.53447640657719775660999853148, −1.73550282431264646745125129550, 2.51903332244233312931291208173, 3.44006597845181513551764232923, 5.86407998785445718987111686513, 6.89944620426769172352808888648, 8.095820178973720601315976554995, 9.108826497477132658211277265415, 10.43466356887440515148271835817, 10.82600781498994100680873042578, 11.67674798406351145090216960613, 13.10800215697424163384851107610

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