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Properties

Label 2-588-196.55-c1-0-36
Degree $2$
Conductor $588$
Sign $-0.901 - 0.433i$
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  + (−0.537 − 1.30i)2-s + (−0.900 − 0.433i)3-s + (−1.42 + 1.40i)4-s + (−1.35 + 2.81i)5-s + (−0.0836 + 1.41i)6-s + (2.57 − 0.617i)7-s + (2.60 + 1.10i)8-s + (0.623 + 0.781i)9-s + (4.40 + 0.261i)10-s + (−1.76 − 1.40i)11-s + (1.89 − 0.648i)12-s + (−3.29 − 2.62i)13-s + (−2.19 − 3.03i)14-s + (2.44 − 1.94i)15-s + (0.0497 − 3.99i)16-s + (−0.732 − 0.167i)17-s + ⋯
L(s)  = 1  + (−0.379 − 0.925i)2-s + (−0.520 − 0.250i)3-s + (−0.711 + 0.702i)4-s + (−0.605 + 1.25i)5-s + (−0.0341 + 0.576i)6-s + (0.972 − 0.233i)7-s + (0.920 + 0.391i)8-s + (0.207 + 0.260i)9-s + (1.39 + 0.0826i)10-s + (−0.532 − 0.424i)11-s + (0.546 − 0.187i)12-s + (−0.912 − 0.727i)13-s + (−0.585 − 0.810i)14-s + (0.630 − 0.502i)15-s + (0.0124 − 0.999i)16-s + (−0.177 − 0.0405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.901 - 0.433i$
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: \((2,\ 588,\ (\ :1/2),\ -0.901 - 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0239139 + 0.104861i\)
\(L(\frac12)\) \(\approx\) \(0.0239139 + 0.104861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.537 + 1.30i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-2.57 + 0.617i)T \)
good5 \( 1 + (1.35 - 2.81i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (1.76 + 1.40i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.29 + 2.62i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.732 + 0.167i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 + (6.10 - 1.39i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.287 + 1.25i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 0.402T + 31T^{2} \)
37 \( 1 + (-0.433 + 1.89i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (0.368 - 0.764i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (5.46 + 11.3i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (7.96 - 9.98i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.85 + 8.13i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (8.17 - 3.93i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (9.90 + 2.26i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 1.76iT - 67T^{2} \)
71 \( 1 + (-9.16 + 2.09i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.14 + 4.10i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 5.36iT - 79T^{2} \)
83 \( 1 + (4.49 + 5.63i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (12.4 - 9.92i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 0.150iT - 97T^{2} \)
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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

−10.67721692284838348307406491809, −9.675564216020543026782185014327, −8.092166612193450044927061517996, −7.84807352734420448605715750912, −6.82374756572355777882242185572, −5.40021013633104653485072524916, −4.32287756664688909984062175842, −3.15654857804560714237427432486, −2.03705653460028891247583733424, −0.07252251871719286092009867599, 1.68188514813610077238466069793, 4.45267825891927924006597507180, 4.63907586716889253403319211559, 5.58489782708576539493378095069, 6.77062892393836666710350428404, 7.890769815003629195088911819437, 8.335390396943619443712009943773, 9.304876429684178206222815581173, 10.09015793495087359993977465851, 11.16466056841954905351139441786

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