Properties

Label 2-588-147.65-c0-0-0
Degree $2$
Conductor $588$
Sign $0.201 - 0.979i$
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  + (0.0747 + 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (0.0931 − 0.116i)13-s + (−0.365 + 0.632i)19-s + (−0.900 + 0.433i)21-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.733 + 1.26i)31-s + (−1.40 − 1.29i)37-s + (0.123 + 0.0841i)39-s + (1.78 − 0.858i)43-s + (−0.733 + 0.680i)49-s + (−0.658 − 0.317i)57-s + (0.326 + 0.302i)61-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (0.0931 − 0.116i)13-s + (−0.365 + 0.632i)19-s + (−0.900 + 0.433i)21-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.733 + 1.26i)31-s + (−1.40 − 1.29i)37-s + (0.123 + 0.0841i)39-s + (1.78 − 0.858i)43-s + (−0.733 + 0.680i)49-s + (−0.658 − 0.317i)57-s + (0.326 + 0.302i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9233181873\)
\(L(\frac12)\) \(\approx\) \(0.9233181873\)
\(L(1)\) 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 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (-0.365 - 0.930i)T \)
good5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-0.326 - 0.302i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
79 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−10.79357082728569519639095271904, −10.41199586215820557577401335107, −9.198264842554699103185159186660, −8.713907011770914578056453870768, −7.80905014141108221262079585424, −6.35076668042073016767483506656, −5.47037861850002388712401376297, −4.61654263122742842290714323482, −3.46481160600584306661912146577, −2.26339410567486130003182245167, 1.23546709850735635284275309449, 2.67402538236873830691499900714, 4.03231416953390946073746124661, 5.24153085192507873973227689592, 6.43128230434516971850366538328, 7.15749228497716721173506811377, 7.943774577113719724398871095920, 8.777048114440234034271017921023, 9.848567644221284712757421374895, 10.96690936815417066079464651638

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