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Properties

Label 2-588-196.167-c1-0-28
Degree $2$
Conductor $588$
Sign $0.970 + 0.243i$
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  + (−0.714 + 1.22i)2-s + (−0.222 − 0.974i)3-s + (−0.979 − 1.74i)4-s + (−0.481 + 0.109i)5-s + (1.34 + 0.424i)6-s + (2.64 − 0.100i)7-s + (2.82 + 0.0502i)8-s + (−0.900 + 0.433i)9-s + (0.209 − 0.666i)10-s + (−1.67 + 3.47i)11-s + (−1.48 + 1.34i)12-s + (2.14 − 4.45i)13-s + (−1.76 + 3.29i)14-s + (0.214 + 0.444i)15-s + (−2.08 + 3.41i)16-s + (−4.80 − 3.83i)17-s + ⋯
L(s)  = 1  + (−0.505 + 0.863i)2-s + (−0.128 − 0.562i)3-s + (−0.489 − 0.871i)4-s + (−0.215 + 0.0491i)5-s + (0.550 + 0.173i)6-s + (0.999 − 0.0381i)7-s + (0.999 + 0.0177i)8-s + (−0.300 + 0.144i)9-s + (0.0663 − 0.210i)10-s + (−0.505 + 1.04i)11-s + (−0.427 + 0.387i)12-s + (0.594 − 1.23i)13-s + (−0.471 + 0.881i)14-s + (0.0553 + 0.114i)15-s + (−0.520 + 0.853i)16-s + (−1.16 − 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.970 + 0.243i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.970 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04390 - 0.128770i\)
\(L(\frac12)\) \(\approx\) \(1.04390 - 0.128770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.714 - 1.22i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (-2.64 + 0.100i)T \)
good5 \( 1 + (0.481 - 0.109i)T + (4.50 - 2.16i)T^{2} \)
11 \( 1 + (1.67 - 3.47i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.14 + 4.45i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (4.80 + 3.83i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 - 4.39T + 19T^{2} \)
23 \( 1 + (-4.21 + 3.36i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-2.16 + 2.71i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 - 7.28T + 31T^{2} \)
37 \( 1 + (-5.96 + 7.47i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-7.01 + 1.60i)T + (36.9 - 17.7i)T^{2} \)
43 \( 1 + (1.75 + 0.400i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.0243 - 0.0117i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-5.85 - 7.34i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-1.17 + 5.13i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (10.6 + 8.51i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + (10.4 - 8.36i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (3.24 + 6.73i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 - 6.48iT - 79T^{2} \)
83 \( 1 + (-0.367 + 0.177i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-5.31 - 11.0i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 - 9.83iT - 97T^{2} \)
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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

−10.68351305206583668995143695685, −9.623472340663610031288663005773, −8.673048352415209359729488528291, −7.67841000537095418363052403734, −7.46855251227215617672893573046, −6.25109792211359427306989691130, −5.22981967327561347802662512467, −4.48352774686515585085327406958, −2.42723009466297758318774861624, −0.847534998349591483443233915590, 1.32487703504139737089616677695, 2.82736717992369228065435558434, 4.05840873772200875914675277621, 4.76588054144362560470665492645, 6.09230697840931907149888602652, 7.50730312157995839753687614369, 8.509685632439848997648380670546, 8.863316402199910748388204680706, 10.00519628781521249226799935103, 10.85736682180413630229777580334

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