Properties

Label 2-588-196.187-c1-0-16
Degree $2$
Conductor $588$
Sign $-0.244 - 0.969i$
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  + (0.427 + 1.34i)2-s + (−0.988 − 0.149i)3-s + (−1.63 + 1.15i)4-s + (0.113 − 0.0447i)5-s + (−0.222 − 1.39i)6-s + (2.30 − 1.29i)7-s + (−2.25 − 1.70i)8-s + (0.955 + 0.294i)9-s + (0.109 + 0.134i)10-s + (0.636 + 2.06i)11-s + (1.78 − 0.897i)12-s + (6.14 + 1.40i)13-s + (2.73 + 2.55i)14-s + (−0.119 + 0.0272i)15-s + (1.33 − 3.76i)16-s + (−1.76 + 2.58i)17-s + ⋯
L(s)  = 1  + (0.302 + 0.953i)2-s + (−0.570 − 0.0860i)3-s + (−0.816 + 0.576i)4-s + (0.0509 − 0.0200i)5-s + (−0.0907 − 0.570i)6-s + (0.872 − 0.488i)7-s + (−0.796 − 0.604i)8-s + (0.318 + 0.0982i)9-s + (0.0344 + 0.0425i)10-s + (0.191 + 0.621i)11-s + (0.515 − 0.258i)12-s + (1.70 + 0.388i)13-s + (0.729 + 0.683i)14-s + (−0.0308 + 0.00703i)15-s + (0.334 − 0.942i)16-s + (−0.428 + 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.244 - 0.969i$
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: \((2,\ 588,\ (\ :1/2),\ -0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880981 + 1.13045i\)
\(L(\frac12)\) \(\approx\) \(0.880981 + 1.13045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.427 - 1.34i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (-2.30 + 1.29i)T \)
good5 \( 1 + (-0.113 + 0.0447i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (-0.636 - 2.06i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-6.14 - 1.40i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (1.76 - 2.58i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (1.73 - 3.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.604 + 0.886i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (-2.36 + 1.14i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-5.04 - 8.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0334 - 0.446i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (9.19 + 7.33i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-2.90 + 2.31i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-8.21 - 7.62i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.0214 + 0.285i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (4.25 - 10.8i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-5.95 - 0.446i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-11.1 + 6.43i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.13 + 2.35i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.271 + 0.292i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (9.88 + 5.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.511 + 2.23i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (2.99 - 9.71i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 - 4.41iT - 97T^{2} \)
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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

−10.92949648196409669184271011512, −10.16302709378854382802995705448, −8.822674896956564626334195249523, −8.254081919674495063110257966415, −7.20075623789912350149387119998, −6.41450434070290881720985558452, −5.58193472033173113352110376148, −4.46847775132950648528721793875, −3.79689964454330123404823867387, −1.48341742214603431125323383853, 0.925237242589801669696363216463, 2.37040852288782603883320589082, 3.75792001985749003583499178754, 4.72489192173841184120754380065, 5.70527865667642038357659242494, 6.41318573451980189141588461589, 8.184346960258492472787517542968, 8.725969767929105237031952450256, 9.801460333888389366768197350328, 10.74303772815514967361982276737

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