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Properties

Label 2-588-12.11-c1-0-12
Degree $2$
Conductor $588$
Sign $0.564 - 0.825i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 0.842i)2-s + (−1.65 − 0.522i)3-s + (0.578 − 1.91i)4-s + 0.499i·5-s + (2.31 − 0.798i)6-s + (0.956 + 2.66i)8-s + (2.45 + 1.72i)9-s + (−0.421 − 0.567i)10-s − 1.39·11-s + (−1.95 + 2.85i)12-s − 2.75·13-s + (0.260 − 0.824i)15-s + (−3.32 − 2.21i)16-s − 5.82i·17-s + (−4.24 + 0.109i)18-s + 2.48i·19-s + ⋯
L(s)  = 1  + (−0.802 + 0.596i)2-s + (−0.953 − 0.301i)3-s + (0.289 − 0.957i)4-s + 0.223i·5-s + (0.945 − 0.326i)6-s + (0.338 + 0.941i)8-s + (0.818 + 0.575i)9-s + (−0.133 − 0.179i)10-s − 0.419·11-s + (−0.564 + 0.825i)12-s − 0.762·13-s + (0.0673 − 0.212i)15-s + (−0.832 − 0.554i)16-s − 1.41i·17-s + (−0.999 + 0.0258i)18-s + 0.569i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560270 + 0.295515i\)
\(L(\frac12)\) \(\approx\) \(0.560270 + 0.295515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.842i)T \)
3 \( 1 + (1.65 + 0.522i)T \)
7 \( 1 \)
good5 \( 1 - 0.499iT - 5T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 - 2.48iT - 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 - 6.32iT - 29T^{2} \)
31 \( 1 - 6.91iT - 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 - 5.74iT - 41T^{2} \)
43 \( 1 - 3.52iT - 43T^{2} \)
47 \( 1 - 5.06T + 47T^{2} \)
53 \( 1 + 5.24iT - 53T^{2} \)
59 \( 1 - 3.15T + 59T^{2} \)
61 \( 1 - 4.90T + 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 - 9.97T + 73T^{2} \)
79 \( 1 - 2.13iT - 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 2.45iT - 89T^{2} \)
97 \( 1 + 0.526T + 97T^{2} \)
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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

−10.76316252491986844164931013263, −9.981721208896295519145582450683, −9.149518120458347859780064153098, −7.971106222603514875053444300660, −7.10628610967939629955393266589, −6.64299038118892289191304231118, −5.32831154098463150229693425358, −4.87835701730318275083804067903, −2.66536741124366375391386997480, −1.00804588519643503859310579725, 0.68881037054859609991264609461, 2.35689056761999694313574828080, 3.87812134736107247882489171806, 4.84950860588833772936152541250, 6.07432066190458533774389797094, 7.08687737364050137101704842937, 7.982771920164255553282251892472, 9.080613315484282028879036660939, 9.797749188400156744470976516332, 10.66847174234874667840587383795

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