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Properties

Label 2-179-179.178-c0-0-0
Degree $2$
Conductor $179$
Sign $1$
Analytic cond. $0.0893326$
Root an. cond. $0.298885$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·3-s + 4-s + 0.618·5-s + 1.61·9-s − 1.61·12-s + 0.618·13-s − 1.00·15-s + 16-s − 1.61·17-s − 1.61·19-s + 0.618·20-s − 0.618·25-s − 27-s − 1.61·29-s + 0.618·31-s + 1.61·36-s − 1.00·39-s + 0.618·43-s + 1.00·45-s + 2·47-s − 1.61·48-s + 49-s + 2.61·51-s + 0.618·52-s + 2.61·57-s − 1.61·59-s − 1.00·60-s + ⋯
L(s)  = 1  − 1.61·3-s + 4-s + 0.618·5-s + 1.61·9-s − 1.61·12-s + 0.618·13-s − 1.00·15-s + 16-s − 1.61·17-s − 1.61·19-s + 0.618·20-s − 0.618·25-s − 27-s − 1.61·29-s + 0.618·31-s + 1.61·36-s − 1.00·39-s + 0.618·43-s + 1.00·45-s + 2·47-s − 1.61·48-s + 49-s + 2.61·51-s + 0.618·52-s + 2.61·57-s − 1.61·59-s − 1.00·60-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(179\)
Sign: $1$
Analytic conductor: \(0.0893326\)
Root analytic conductor: \(0.298885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (178, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 179,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5633125595\)
\(L(\frac12)\) \(\approx\) \(0.5633125595\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 - T \)
good2 \( 1 - T^{2} \)
3 \( 1 + 1.61T + T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 0.618T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.66621200984282707902061978356, −11.76904835393458998002817039838, −10.81180595749901060465169721024, −10.59782383628861066854338835634, −9.034986564804756347768170299020, −7.34624348417058841171468547738, −6.23151086383162471560444554494, −5.92127799396673574655226895630, −4.35201337284568871813129465330, −2.01597167972204833908759072878, 2.01597167972204833908759072878, 4.35201337284568871813129465330, 5.92127799396673574655226895630, 6.23151086383162471560444554494, 7.34624348417058841171468547738, 9.034986564804756347768170299020, 10.59782383628861066854338835634, 10.81180595749901060465169721024, 11.76904835393458998002817039838, 12.66621200984282707902061978356

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