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Properties

Label 2-588-49.22-c1-0-3
Degree $2$
Conductor $588$
Sign $-0.286 - 0.958i$
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  + (−0.222 + 0.974i)3-s + (−0.797 + 3.49i)5-s + (2.64 − 0.172i)7-s + (−0.900 − 0.433i)9-s + (4.83 − 2.32i)11-s + (−1.75 + 0.846i)13-s + (−3.23 − 1.55i)15-s + (4.48 + 5.62i)17-s − 1.92·19-s + (−0.419 + 2.61i)21-s + (−5.00 + 6.27i)23-s + (−7.07 − 3.40i)25-s + (0.623 − 0.781i)27-s + (−3.37 − 4.23i)29-s − 5.12·31-s + ⋯
L(s)  = 1  + (−0.128 + 0.562i)3-s + (−0.356 + 1.56i)5-s + (0.997 − 0.0650i)7-s + (−0.300 − 0.144i)9-s + (1.45 − 0.701i)11-s + (−0.487 + 0.234i)13-s + (−0.834 − 0.401i)15-s + (1.08 + 1.36i)17-s − 0.442·19-s + (−0.0915 + 0.570i)21-s + (−1.04 + 1.30i)23-s + (−1.41 − 0.681i)25-s + (0.119 − 0.150i)27-s + (−0.627 − 0.786i)29-s − 0.921·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.286 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856001 + 1.14941i\)
\(L(\frac12)\) \(\approx\) \(0.856001 + 1.14941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-2.64 + 0.172i)T \)
good5 \( 1 + (0.797 - 3.49i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-4.83 + 2.32i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.75 - 0.846i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-4.48 - 5.62i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 1.92T + 19T^{2} \)
23 \( 1 + (5.00 - 6.27i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (3.37 + 4.23i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + (4.02 + 5.04i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (1.73 - 7.59i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-1.27 - 5.60i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-10.5 + 5.07i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-4.38 + 5.50i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.402 - 1.76i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (8.60 + 10.7i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + (2.53 - 3.18i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.28 + 0.619i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-1.89 - 0.912i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-4.98 - 2.40i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 0.407T + 97T^{2} \)
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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

−11.03332528180743930999567188330, −10.24186916037994204312248627594, −9.361239118745341169712911823938, −8.185615641249558862020822309781, −7.45365333946881231412640773363, −6.36863585072160498349641373828, −5.59954652168724035860661119084, −3.96126236236353898495444238647, −3.59060238837790608502249534036, −1.89625291418518022183102289198, 0.888137506072688059419854712952, 1.99148724119964656362798994114, 4.01418382605736798952852811267, 4.86022800930255895812867848327, 5.63102696689582076928118779122, 7.07910108367216340233322497009, 7.75791573572183630562218981565, 8.759846422683893266245068365028, 9.218000857101077259628354588812, 10.46893471848604708642997051314

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