Properties

Label 2-588-196.187-c1-0-4
Degree $2$
Conductor $588$
Sign $-0.124 - 0.992i$
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  + (−1.30 + 0.549i)2-s + (−0.988 − 0.149i)3-s + (1.39 − 1.43i)4-s + (−2.67 + 1.05i)5-s + (1.37 − 0.348i)6-s + (−0.927 − 2.47i)7-s + (−1.03 + 2.63i)8-s + (0.955 + 0.294i)9-s + (2.90 − 2.83i)10-s + (−1.21 − 3.93i)11-s + (−1.59 + 1.20i)12-s + (4.15 + 0.948i)13-s + (2.57 + 2.71i)14-s + (2.80 − 0.639i)15-s + (−0.0991 − 3.99i)16-s + (−1.41 + 2.07i)17-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (−0.570 − 0.0860i)3-s + (0.698 − 0.715i)4-s + (−1.19 + 0.469i)5-s + (0.559 − 0.142i)6-s + (−0.350 − 0.936i)7-s + (−0.365 + 0.930i)8-s + (0.318 + 0.0982i)9-s + (0.920 − 0.897i)10-s + (−0.365 − 1.18i)11-s + (−0.460 + 0.348i)12-s + (1.15 + 0.262i)13-s + (0.686 + 0.726i)14-s + (0.723 − 0.165i)15-s + (−0.0247 − 0.999i)16-s + (−0.342 + 0.502i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.124 - 0.992i$
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: \((2,\ 588,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241027 + 0.273298i\)
\(L(\frac12)\) \(\approx\) \(0.241027 + 0.273298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.549i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.927 + 2.47i)T \)
good5 \( 1 + (2.67 - 1.05i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (1.21 + 3.93i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-4.15 - 0.948i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (1.41 - 2.07i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (1.77 - 3.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0806 + 0.118i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (1.88 - 0.905i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-2.09 - 3.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.213 - 2.84i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-5.74 - 4.58i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-0.530 + 0.422i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.03 - 3.74i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.987 - 13.1i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (0.259 - 0.661i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (8.24 + 0.617i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (13.2 - 7.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.00 + 12.4i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-8.63 - 9.30i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-0.803 - 0.463i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.42 - 15.0i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-4.14 + 13.4i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 + 1.93iT - 97T^{2} \)
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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

−10.87391940823872711300127965834, −10.37302270266965707560680959513, −9.013599198798047570761593263412, −8.112061797823225156911589851846, −7.51504699865793258813499446877, −6.49592886491141503682831594979, −5.92424418017477004495064803845, −4.27010853010265346324967524593, −3.23478167397751177179063912692, −1.07709204594009389902807317733, 0.36307312004229800848581699242, 2.22519980550406112752422249530, 3.65589336390072172935826807409, 4.71058691516972008006198003228, 6.05582749977515526943765820777, 7.10044872518084991265908427539, 7.934716198921215129390536365510, 8.807016443026779739591183887681, 9.481987805937519517897178661724, 10.55989931959110970942866460478

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