Properties

Degree $2$
Conductor $588$
Sign $-0.991 - 0.126i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−1 + 1.73i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 4·13-s + 1.99·15-s + (−3 − 5.19i)17-s + (−4 + 6.92i)19-s + (3 − 5.19i)23-s + (0.500 + 0.866i)25-s + 0.999·27-s − 10·29-s + (−2 − 3.46i)31-s + (−0.999 + 1.73i)33-s + (−3 + 5.19i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 1.10·13-s + 0.516·15-s + (−0.727 − 1.26i)17-s + (−0.917 + 1.58i)19-s + (0.625 − 1.08i)23-s + (0.100 + 0.173i)25-s + 0.192·27-s − 1.85·29-s + (−0.359 − 0.622i)31-s + (−0.174 + 0.301i)33-s + (−0.493 + 0.854i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Motivic weight: \(1\)
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−10.44762962121687293381199143183, −9.410304692521820771554835525408, −8.290665314179061344772246094631, −7.41671156228122135649852881403, −6.79172440819309870546315464275, −5.74190479357973473147643879227, −4.63821733801954034466714438835, −3.28601497984755703972909028945, −2.16795692825738455667006861215, 0, 2.10996305030728947539194238780, 3.75178103956797459541164325423, 4.71287893562689173074985731068, 5.34430176953819470610031684652, 6.73046060607173898539584677331, 7.58623274027051951656007176582, 8.765490601545603914125011455773, 9.252782538482654723176739408920, 10.36758497730667856935644977680

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