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Properties

Label 2-179-179.178-c0-0-1
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  + 0.618·3-s + 4-s − 1.61·5-s − 0.618·9-s + 0.618·12-s − 1.61·13-s − 1.00·15-s + 16-s + 0.618·17-s + 0.618·19-s − 1.61·20-s + 1.61·25-s − 27-s + 0.618·29-s − 1.61·31-s − 0.618·36-s − 1.00·39-s − 1.61·43-s + 0.999·45-s + 2·47-s + 0.618·48-s + 49-s + 0.381·51-s − 1.61·52-s + 0.381·57-s + 0.618·59-s − 1.00·60-s + ⋯
L(s)  = 1  + 0.618·3-s + 4-s − 1.61·5-s − 0.618·9-s + 0.618·12-s − 1.61·13-s − 1.00·15-s + 16-s + 0.618·17-s + 0.618·19-s − 1.61·20-s + 1.61·25-s − 27-s + 0.618·29-s − 1.61·31-s − 0.618·36-s − 1.00·39-s − 1.61·43-s + 0.999·45-s + 2·47-s + 0.618·48-s + 49-s + 0.381·51-s − 1.61·52-s + 0.381·57-s + 0.618·59-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

\(L(\frac{1}{2})\) \(\approx\) \(0.7386413735\)
\(L(\frac12)\) \(\approx\) \(0.7386413735\)
\(L(1)\) 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 \( 1 - 0.618T + T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.618T + T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.61T + T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
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

−12.51919690578070020222115761522, −11.90142378760168376774359235858, −11.21457750147824232248503585176, −10.00257778988186122508545792801, −8.614009732536384041379102799466, −7.61568450637734517093989314558, −7.16148182035105530282500226846, −5.36727342655659964311710483246, −3.70927146165226780110215121662, −2.65584139527623233165233402303, 2.65584139527623233165233402303, 3.70927146165226780110215121662, 5.36727342655659964311710483246, 7.16148182035105530282500226846, 7.61568450637734517093989314558, 8.614009732536384041379102799466, 10.00257778988186122508545792801, 11.21457750147824232248503585176, 11.90142378760168376774359235858, 12.51919690578070020222115761522

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