Properties

Label 2-588-196.187-c1-0-54
Degree $2$
Conductor $588$
Sign $-0.817 + 0.575i$
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  + (0.978 − 1.02i)2-s + (−0.988 − 0.149i)3-s + (−0.0843 − 1.99i)4-s + (2.70 − 1.06i)5-s + (−1.11 + 0.863i)6-s + (−2.19 − 1.47i)7-s + (−2.12 − 1.86i)8-s + (0.955 + 0.294i)9-s + (1.56 − 3.80i)10-s + (−0.192 − 0.622i)11-s + (−0.214 + 1.98i)12-s + (−1.67 − 0.381i)13-s + (−3.65 + 0.800i)14-s + (−2.83 + 0.647i)15-s + (−3.98 + 0.336i)16-s + (1.43 − 2.10i)17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (−0.570 − 0.0860i)3-s + (−0.0421 − 0.999i)4-s + (1.21 − 0.475i)5-s + (−0.457 + 0.352i)6-s + (−0.830 − 0.557i)7-s + (−0.750 − 0.660i)8-s + (0.318 + 0.0982i)9-s + (0.495 − 1.20i)10-s + (−0.0579 − 0.187i)11-s + (−0.0619 + 0.574i)12-s + (−0.463 − 0.105i)13-s + (−0.976 + 0.214i)14-s + (−0.732 + 0.167i)15-s + (−0.996 + 0.0842i)16-s + (0.347 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524957 - 1.65869i\)
\(L(\frac12)\) \(\approx\) \(0.524957 - 1.65869i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.978 + 1.02i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (2.19 + 1.47i)T \)
good5 \( 1 + (-2.70 + 1.06i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (0.192 + 0.622i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (1.67 + 0.381i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.43 + 2.10i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (-0.182 + 0.316i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.51 - 3.69i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-3.73 + 1.79i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (3.29 + 5.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0988 - 1.31i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (5.90 + 4.71i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-3.72 + 2.96i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-1.94 - 1.80i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.210 - 2.81i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (3.17 - 8.09i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-3.20 - 0.239i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-9.01 + 5.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.67 + 11.7i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-6.46 - 6.96i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-13.4 - 7.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.33 + 5.85i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-4.82 + 15.6i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−10.33273674773421733802041635470, −9.724711089770522515766516156917, −9.127379591366137954828039042977, −7.35527862798821795865895280357, −6.34894614024075333713689118723, −5.61083911051773591366380952864, −4.85155289689481711884699199506, −3.55783818063124545043774290460, −2.26353392127596762784693985342, −0.836424680185339894952094022095, 2.30910889033759812925106027844, 3.40936230049379293966335783886, 4.90103463487171244554239904464, 5.64098088757172834259705061651, 6.48772378755570980126435717909, 6.92342886927741154759861170892, 8.344470681553849060502664332490, 9.381551197323126950812087268178, 10.09550259464509026570897413219, 11.04940886516515447481309595895

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