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Properties

Label 2-588-4.3-c2-0-62
Degree $2$
Conductor $588$
Sign $0.364 + 0.931i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 + 0.371i)2-s + 1.73i·3-s + (3.72 − 1.45i)4-s + 4.44·5-s + (−0.642 − 3.40i)6-s + (−6.77 + 4.24i)8-s − 2.99·9-s + (−8.73 + 1.65i)10-s − 14.7i·11-s + (2.52 + 6.45i)12-s − 0.580·13-s + 7.70i·15-s + (11.7 − 10.8i)16-s − 1.84·17-s + (5.89 − 1.11i)18-s − 27.1i·19-s + ⋯
L(s)  = 1  + (−0.982 + 0.185i)2-s + 0.577i·3-s + (0.931 − 0.364i)4-s + 0.889·5-s + (−0.107 − 0.567i)6-s + (−0.847 + 0.531i)8-s − 0.333·9-s + (−0.873 + 0.165i)10-s − 1.34i·11-s + (0.210 + 0.537i)12-s − 0.0446·13-s + 0.513i·15-s + (0.733 − 0.679i)16-s − 0.108·17-s + (0.327 − 0.0618i)18-s − 1.42i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.364 + 0.931i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9324455135\)
\(L(\frac12)\) \(\approx\) \(0.9324455135\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 - 0.371i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 4.44T + 25T^{2} \)
11 \( 1 + 14.7iT - 121T^{2} \)
13 \( 1 + 0.580T + 169T^{2} \)
17 \( 1 + 1.84T + 289T^{2} \)
19 \( 1 + 27.1iT - 361T^{2} \)
23 \( 1 + 32.0iT - 529T^{2} \)
29 \( 1 + 52.0T + 841T^{2} \)
31 \( 1 - 5.90iT - 961T^{2} \)
37 \( 1 - 29.9T + 1.36e3T^{2} \)
41 \( 1 + 34.8T + 1.68e3T^{2} \)
43 \( 1 - 63.2iT - 1.84e3T^{2} \)
47 \( 1 + 58.0iT - 2.20e3T^{2} \)
53 \( 1 + 24.1T + 2.80e3T^{2} \)
59 \( 1 + 64.9iT - 3.48e3T^{2} \)
61 \( 1 + 44.3T + 3.72e3T^{2} \)
67 \( 1 + 42.3iT - 4.48e3T^{2} \)
71 \( 1 - 33.6iT - 5.04e3T^{2} \)
73 \( 1 - 130.T + 5.32e3T^{2} \)
79 \( 1 - 48.7iT - 6.24e3T^{2} \)
83 \( 1 + 137. iT - 6.88e3T^{2} \)
89 \( 1 - 116.T + 7.92e3T^{2} \)
97 \( 1 - 133.T + 9.40e3T^{2} \)
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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

−10.18998039226094484784747079759, −9.354406405386484982663186816559, −8.820072694618700098849944666537, −7.900064124410035701754549103804, −6.61728901422641978118572189860, −5.95112820899504005252249586577, −4.96629997015053466255875185269, −3.26782284982406678206726725904, −2.14819520245156124293840195965, −0.46279979615082776705454071851, 1.55858332867109110588795407921, 2.15956074576036546771044498456, 3.70390378538773047700636268668, 5.48586465876518551714962618297, 6.28202817261308231505250423469, 7.38178465604657181431966729106, 7.82928361842066316182355598400, 9.165023465516642650423087531761, 9.676976040754240010304872852890, 10.41245095788848293461337946997

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