Properties

Label 2-588-196.187-c1-0-52
Degree $2$
Conductor $588$
Sign $-0.119 + 0.992i$
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  + (0.789 − 1.17i)2-s + (0.988 + 0.149i)3-s + (−0.753 − 1.85i)4-s + (2.68 − 1.05i)5-s + (0.955 − 1.04i)6-s + (0.891 − 2.49i)7-s + (−2.76 − 0.579i)8-s + (0.955 + 0.294i)9-s + (0.884 − 3.98i)10-s + (0.268 + 0.869i)11-s + (−0.468 − 1.94i)12-s + (2.28 + 0.520i)13-s + (−2.21 − 3.01i)14-s + (2.81 − 0.642i)15-s + (−2.86 + 2.79i)16-s + (−3.81 + 5.59i)17-s + ⋯
L(s)  = 1  + (0.558 − 0.829i)2-s + (0.570 + 0.0860i)3-s + (−0.376 − 0.926i)4-s + (1.20 − 0.471i)5-s + (0.390 − 0.425i)6-s + (0.337 − 0.941i)7-s + (−0.978 − 0.204i)8-s + (0.318 + 0.0982i)9-s + (0.279 − 1.26i)10-s + (0.0808 + 0.262i)11-s + (−0.135 − 0.561i)12-s + (0.633 + 0.144i)13-s + (−0.592 − 0.805i)14-s + (0.727 − 0.165i)15-s + (−0.716 + 0.697i)16-s + (−0.925 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.119 + 0.992i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78926 - 2.01743i\)
\(L(\frac12)\) \(\approx\) \(1.78926 - 2.01743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.789 + 1.17i)T \)
3 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.891 + 2.49i)T \)
good5 \( 1 + (-2.68 + 1.05i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (-0.268 - 0.869i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-2.28 - 0.520i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (3.81 - 5.59i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (3.81 - 6.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.79 + 4.10i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-5.68 + 2.73i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (1.62 + 2.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.408 + 5.45i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-5.53 - 4.41i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (0.978 - 0.780i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.33 + 2.16i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.936 - 12.4i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-3.10 + 7.92i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-5.72 - 0.428i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (2.42 - 1.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.50 - 3.11i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.814 - 0.878i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (13.4 + 7.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.65 - 7.23i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (3.16 - 10.2i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−10.35772876096715085166875378765, −9.914307930353942145314595756405, −8.851440255206049029200210889897, −8.132433287286425086155341905521, −6.46209343748232753413899984317, −5.85316408078580185064465851861, −4.40099537132646970161795973204, −3.95050334187821565922579958883, −2.24040336430439550607474631716, −1.45982235862405564433802556191, 2.26348577662706951937021473475, 3.04073675675934114692927456990, 4.61162191329914633496618043155, 5.52222725490823952324556908244, 6.43781412274205007525993785178, 7.08187085165574451458160238635, 8.444952792302034635210425114743, 8.926143220953005380637107244483, 9.716531744567461178843776848530, 11.04635713681835742589537754977

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