Properties

Label 2-588-196.27-c1-0-14
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: $\chi_{588} (223, \cdot )$
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

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

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