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Properties

Label 2-588-147.53-c0-0-0
Degree $2$
Conductor $588$
Sign $0.871 + 0.490i$
Analytic cond. $0.293450$
Root an. cond. $0.541710$
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  + (−0.733 − 0.680i)3-s + (0.826 + 0.563i)7-s + (0.0747 + 0.997i)9-s + (1.32 − 0.636i)13-s + (−0.826 − 1.43i)19-s + (−0.222 − 0.974i)21-s + (0.826 − 0.563i)25-s + (0.623 − 0.781i)27-s + (−0.365 + 0.632i)31-s + (−0.722 + 1.84i)37-s + (−1.40 − 0.432i)39-s + (−0.0332 − 0.145i)43-s + (0.365 + 0.930i)49-s + (−0.367 + 1.61i)57-s + (0.455 − 1.16i)61-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)3-s + (0.826 + 0.563i)7-s + (0.0747 + 0.997i)9-s + (1.32 − 0.636i)13-s + (−0.826 − 1.43i)19-s + (−0.222 − 0.974i)21-s + (0.826 − 0.563i)25-s + (0.623 − 0.781i)27-s + (−0.365 + 0.632i)31-s + (−0.722 + 1.84i)37-s + (−1.40 − 0.432i)39-s + (−0.0332 − 0.145i)43-s + (0.365 + 0.930i)49-s + (−0.367 + 1.61i)57-s + (0.455 − 1.16i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8125831672\)
\(L(\frac12)\) \(\approx\) \(0.8125831672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (-0.826 - 0.563i)T \)
good5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-1.32 + 0.636i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (-0.455 + 1.16i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−11.02010075548298563723948983226, −10.28832913895149030494758612534, −8.666909574551318192012537603553, −8.384674358896155912796513765542, −7.12420332631262603013942076682, −6.31544652676262146991686634546, −5.37737642715632960602250969410, −4.53559832269031322605302017410, −2.79054051921709119995551940344, −1.38529926558741575824058485651, 1.54830538513620671623865839789, 3.69269407874133238887112002482, 4.30775460598029860799587350196, 5.48068752052555652537479533080, 6.28527043785033269215236432968, 7.35779744652434649894763564426, 8.492331343020060675186011524153, 9.240310357034639558975934851093, 10.46958859416177443387350263634, 10.82441330119665726532307599614

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