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Properties

Label 2-588-196.55-c1-0-8
Degree $2$
Conductor $588$
Sign $0.917 + 0.398i$
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  + (0.0809 − 1.41i)2-s + (−0.900 − 0.433i)3-s + (−1.98 − 0.228i)4-s + (−0.881 + 1.83i)5-s + (−0.685 + 1.23i)6-s + (−1.70 − 2.02i)7-s + (−0.483 + 2.78i)8-s + (0.623 + 0.781i)9-s + (2.51 + 1.39i)10-s + (3.11 + 2.48i)11-s + (1.69 + 1.06i)12-s + (1.46 + 1.16i)13-s + (−2.99 + 2.24i)14-s + (1.58 − 1.26i)15-s + (3.89 + 0.908i)16-s + (1.45 + 0.333i)17-s + ⋯
L(s)  = 1  + (0.0572 − 0.998i)2-s + (−0.520 − 0.250i)3-s + (−0.993 − 0.114i)4-s + (−0.394 + 0.819i)5-s + (−0.279 + 0.504i)6-s + (−0.645 − 0.763i)7-s + (−0.170 + 0.985i)8-s + (0.207 + 0.260i)9-s + (0.795 + 0.440i)10-s + (0.940 + 0.750i)11-s + (0.488 + 0.308i)12-s + (0.405 + 0.323i)13-s + (−0.799 + 0.600i)14-s + (0.410 − 0.327i)15-s + (0.973 + 0.227i)16-s + (0.353 + 0.0807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928084 - 0.192887i\)
\(L(\frac12)\) \(\approx\) \(0.928084 - 0.192887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0809 + 1.41i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (1.70 + 2.02i)T \)
good5 \( 1 + (0.881 - 1.83i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-3.11 - 2.48i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.46 - 1.16i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.45 - 0.333i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
23 \( 1 + (-5.98 + 1.36i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.581 + 2.54i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + (1.48 - 6.51i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.31 + 8.95i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-5.51 - 11.4i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (6.29 - 7.88i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.0266 + 0.116i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-4.65 + 2.24i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.32 + 1.21i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 7.66iT - 67T^{2} \)
71 \( 1 + (-10.2 + 2.34i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.170 - 0.136i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 2.43iT - 79T^{2} \)
83 \( 1 + (-4.29 - 5.38i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (6.92 - 5.52i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 - 14.2iT - 97T^{2} \)
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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

−10.84594320809810933258480601141, −10.00781311868882248242123431799, −9.223002548768158713705139667728, −7.997942040285588380840705990423, −6.86054590861377494322621189674, −6.29360684842110352725743662780, −4.65322741202528927508313264748, −3.90299369443227217209490965781, −2.77631918516994491669792137359, −1.15880827676115739142447552656, 0.71630364232105756476384033837, 3.37237984133616493202941220286, 4.39761728042330198056133783299, 5.38921140291673911883940052118, 6.16209535149564741394239157556, 6.91967325808715000632160666265, 8.347621207684409360022653739828, 8.794650106828630949480440677168, 9.541389296815144663758194559664, 10.68663425451122871499948872843

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