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Properties

Label 2-588-196.55-c1-0-11
Degree $2$
Conductor $588$
Sign $-0.517 - 0.855i$
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  + (0.652 + 1.25i)2-s + (−0.900 − 0.433i)3-s + (−1.14 + 1.63i)4-s + (−0.0842 + 0.174i)5-s + (−0.0434 − 1.41i)6-s + (2.61 + 0.378i)7-s + (−2.80 − 0.373i)8-s + (0.623 + 0.781i)9-s + (−0.274 + 0.00843i)10-s + (1.26 + 1.01i)11-s + (1.74 − 0.976i)12-s + (3.76 + 2.99i)13-s + (1.23 + 3.53i)14-s + (0.151 − 0.121i)15-s + (−1.36 − 3.76i)16-s + (1.09 + 0.249i)17-s + ⋯
L(s)  = 1  + (0.461 + 0.887i)2-s + (−0.520 − 0.250i)3-s + (−0.574 + 0.818i)4-s + (−0.0376 + 0.0782i)5-s + (−0.0177 − 0.577i)6-s + (0.989 + 0.142i)7-s + (−0.991 − 0.131i)8-s + (0.207 + 0.260i)9-s + (−0.0867 + 0.00266i)10-s + (0.382 + 0.304i)11-s + (0.503 − 0.281i)12-s + (1.04 + 0.831i)13-s + (0.329 + 0.944i)14-s + (0.0391 − 0.0312i)15-s + (−0.340 − 0.940i)16-s + (0.265 + 0.0605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.517 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742855 + 1.31725i\)
\(L(\frac12)\) \(\approx\) \(0.742855 + 1.31725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.652 - 1.25i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-2.61 - 0.378i)T \)
good5 \( 1 + (0.0842 - 0.174i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.26 - 1.01i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.76 - 2.99i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.09 - 0.249i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 6.56T + 19T^{2} \)
23 \( 1 + (5.93 - 1.35i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.33 - 10.2i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 + (1.73 - 7.60i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-1.79 + 3.72i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.62 + 3.36i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.502 + 0.629i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (3.09 + 13.5i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-11.7 + 5.65i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-9.88 - 2.25i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 9.60iT - 67T^{2} \)
71 \( 1 + (10.0 - 2.29i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.14 + 1.71i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + (3.09 + 3.88i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.41 - 1.13i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−11.27668564746813316913342173227, −10.16393303456221175348263812190, −8.745758951685996419948984945845, −8.386285894444683929449141400243, −7.15128300201204328687265377375, −6.52304906392516493315228865686, −5.54780065009114409323323521382, −4.62598570434876315064438089053, −3.70977922522756458670177858336, −1.75484239321350243591198400548, 0.847927126675520716155161152092, 2.32391386406131591993017429289, 3.93238671873695593007773654019, 4.47415174429805483601741629635, 5.73880059872047564122660416458, 6.28836887617246724467823747362, 8.050691373529062331822717670830, 8.670651201595555712854758947000, 9.917235601750780643816058708089, 10.64163724866581040321531659490

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