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Properties

Label 2-588-196.139-c1-0-16
Degree $2$
Conductor $588$
Sign $0.598 - 0.801i$
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  + (−1.37 − 0.345i)2-s + (0.900 − 0.433i)3-s + (1.76 + 0.946i)4-s + (1.63 + 3.39i)5-s + (−1.38 + 0.284i)6-s + (−0.364 + 2.62i)7-s + (−2.08 − 1.90i)8-s + (0.623 − 0.781i)9-s + (−1.07 − 5.21i)10-s + (3.48 − 2.77i)11-s + (1.99 + 0.0883i)12-s + (0.0601 − 0.0479i)13-s + (1.40 − 3.46i)14-s + (2.94 + 2.34i)15-s + (2.20 + 3.33i)16-s + (−5.54 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.969 − 0.244i)2-s + (0.520 − 0.250i)3-s + (0.880 + 0.473i)4-s + (0.730 + 1.51i)5-s + (−0.565 + 0.115i)6-s + (−0.137 + 0.990i)7-s + (−0.738 − 0.673i)8-s + (0.207 − 0.260i)9-s + (−0.338 − 1.65i)10-s + (1.05 − 0.837i)11-s + (0.576 + 0.0255i)12-s + (0.0166 − 0.0132i)13-s + (0.375 − 0.926i)14-s + (0.760 + 0.606i)15-s + (0.552 + 0.833i)16-s + (−1.34 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16530 + 0.584240i\)
\(L(\frac12)\) \(\approx\) \(1.16530 + 0.584240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.345i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.364 - 2.62i)T \)
good5 \( 1 + (-1.63 - 3.39i)T + (-3.11 + 3.90i)T^{2} \)
11 \( 1 + (-3.48 + 2.77i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.0601 + 0.0479i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (5.54 - 1.26i)T + (15.3 - 7.37i)T^{2} \)
19 \( 1 - 8.34T + 19T^{2} \)
23 \( 1 + (6.27 + 1.43i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (-1.35 - 5.94i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 0.0893T + 31T^{2} \)
37 \( 1 + (-1.67 - 7.33i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (2.59 + 5.39i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.87 + 3.89i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.265 - 0.332i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.25 - 5.51i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-3.16 - 1.52i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-9.24 + 2.10i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 0.257iT - 67T^{2} \)
71 \( 1 + (8.57 + 1.95i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.62 + 1.29i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + 3.00iT - 79T^{2} \)
83 \( 1 + (-9.36 + 11.7i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-6.00 - 4.79i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + 1.98iT - 97T^{2} \)
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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

−10.68578686863450915504440280557, −9.828923720798987305178741394654, −9.124505918734394914109512805157, −8.422301116428989754813799652974, −7.22649791459300341221362030177, −6.51290705594504698905866446104, −5.86070236171511692031657818750, −3.50961584596200075334276485376, −2.77080693478229800422664366606, −1.76820473599849747706744994505, 1.01134029255057734715676511425, 2.09436226315969052786405117374, 4.00081807274512553748758464686, 4.98075962168937285376520931493, 6.18418069002305056822579646594, 7.22452490813742925300209713648, 8.068445124356535133135706187984, 9.013387827729713167734358646117, 9.725605719366393605663370499530, 9.865681011266421809254964001267

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