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Properties

Label 2-588-28.27-c1-0-5
Degree $2$
Conductor $588$
Sign $0.990 - 0.134i$
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  + (−0.988 − 1.01i)2-s − 3-s + (−0.0442 + 1.99i)4-s − 1.12i·5-s + (0.988 + 1.01i)6-s + (2.06 − 1.93i)8-s + 9-s + (−1.14 + 1.11i)10-s + 1.35i·11-s + (0.0442 − 1.99i)12-s + 5.58i·13-s + 1.12i·15-s + (−3.99 − 0.177i)16-s − 3.97i·17-s + (−0.988 − 1.01i)18-s − 6.14·19-s + ⋯
L(s)  = 1  + (−0.699 − 0.714i)2-s − 0.577·3-s + (−0.0221 + 0.999i)4-s − 0.504i·5-s + (0.403 + 0.412i)6-s + (0.730 − 0.683i)8-s + 0.333·9-s + (−0.360 + 0.353i)10-s + 0.407i·11-s + (0.0127 − 0.577i)12-s + 1.54i·13-s + 0.291i·15-s + (−0.999 − 0.0442i)16-s − 0.963i·17-s + (−0.233 − 0.238i)18-s − 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743812 + 0.0503048i\)
\(L(\frac12)\) \(\approx\) \(0.743812 + 0.0503048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 + 1.01i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 1.12iT - 5T^{2} \)
11 \( 1 - 1.35iT - 11T^{2} \)
13 \( 1 - 5.58iT - 13T^{2} \)
17 \( 1 + 3.97iT - 17T^{2} \)
19 \( 1 + 6.14T + 19T^{2} \)
23 \( 1 - 6.61iT - 23T^{2} \)
29 \( 1 - 8.55T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 - 6.32T + 37T^{2} \)
41 \( 1 - 0.149iT - 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 - 0.733T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 9.11iT - 61T^{2} \)
67 \( 1 + 0.234iT - 67T^{2} \)
71 \( 1 + 8.74iT - 71T^{2} \)
73 \( 1 - 1.76iT - 73T^{2} \)
79 \( 1 + 8.40iT - 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 - 7.92iT - 89T^{2} \)
97 \( 1 - 6.23iT - 97T^{2} \)
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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

−10.75674242280221109372588391353, −9.785580858342947089286143332431, −9.172363399679461104391081246689, −8.297168622527189096830848004448, −7.17500659250840484223474665474, −6.42174107773697990840552001550, −4.80773593413775398928617465942, −4.17839325734977092385002703379, −2.52784460108444503538390616772, −1.19547476755475920160614407091, 0.67402527198949072103493791312, 2.57014348431682482815117370003, 4.31093667440588856669491897680, 5.47677130755153509099044409923, 6.30626715552299942976240741641, 6.91710887562439147423427824743, 8.263236944571290111212622451415, 8.516973620035802913646800542662, 10.11756613296232880561050015855, 10.48482179237241919464114020579

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