Properties

Label 2-588-196.27-c1-0-2
Degree $2$
Conductor $588$
Sign $-0.746 - 0.665i$
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  + (−1.40 − 0.168i)2-s + (0.222 − 0.974i)3-s + (1.94 + 0.472i)4-s + (−1.38 − 0.316i)5-s + (−0.476 + 1.33i)6-s + (0.327 + 2.62i)7-s + (−2.64 − 0.990i)8-s + (−0.900 − 0.433i)9-s + (1.89 + 0.677i)10-s + (−0.736 − 1.52i)11-s + (0.893 − 1.78i)12-s + (0.572 + 1.18i)13-s + (−0.0173 − 3.74i)14-s + (−0.616 + 1.28i)15-s + (3.55 + 1.83i)16-s + (−1.94 + 1.54i)17-s + ⋯
L(s)  = 1  + (−0.992 − 0.119i)2-s + (0.128 − 0.562i)3-s + (0.971 + 0.236i)4-s + (−0.620 − 0.141i)5-s + (−0.194 + 0.543i)6-s + (0.123 + 0.992i)7-s + (−0.936 − 0.350i)8-s + (−0.300 − 0.144i)9-s + (0.598 + 0.214i)10-s + (−0.221 − 0.460i)11-s + (0.257 − 0.516i)12-s + (0.158 + 0.329i)13-s + (−0.00463 − 0.999i)14-s + (−0.159 + 0.330i)15-s + (0.888 + 0.459i)16-s + (−0.471 + 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.746 - 0.665i$
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: \((2,\ 588,\ (\ :1/2),\ -0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0468996 + 0.123096i\)
\(L(\frac12)\) \(\approx\) \(0.0468996 + 0.123096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.168i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-0.327 - 2.62i)T \)
good5 \( 1 + (1.38 + 0.316i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (0.736 + 1.52i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.572 - 1.18i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.94 - 1.54i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 7.65T + 19T^{2} \)
23 \( 1 + (6.25 + 4.98i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-5.04 - 6.32i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.568T + 31T^{2} \)
37 \( 1 + (-2.14 - 2.68i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.860 - 0.196i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (3.14 - 0.716i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (5.38 - 2.59i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (3.41 - 4.27i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-2.27 - 9.94i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.303 - 0.241i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 - 9.48iT - 67T^{2} \)
71 \( 1 + (11.7 + 9.35i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.03 + 2.14i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 17.1iT - 79T^{2} \)
83 \( 1 + (2.34 + 1.12i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (5.14 - 10.6i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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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

−10.97923593231507095631115843219, −10.20847526381219905285171653174, −8.861248157423445119858496363731, −8.499693378178392353211469115475, −7.85192099202674085854374603545, −6.54258942585023318684211248976, −6.05539729999503547866144121428, −4.34025881761105590177998871253, −2.84854215038890543937093017405, −1.82995053062711461845668587355, 0.092904494538410450280319114671, 2.09678689269703184788025151192, 3.60726616659470089868948547815, 4.54791990164907884998669090088, 6.03005806689215702341258106885, 7.01412863201775383903232389020, 7.931905920140530759800008765053, 8.413335036829231163086825158592, 9.750410815442365156101909599426, 10.13073385062984109222171133217

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