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Properties

Label 2-588-12.11-c1-0-32
Degree $2$
Conductor $588$
Sign $0.998 + 0.0547i$
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  + (0.446 + 1.34i)2-s + (−1.32 − 1.11i)3-s + (−1.60 + 1.19i)4-s − 0.803i·5-s + (0.899 − 2.27i)6-s + (−2.32 − 1.61i)8-s + (0.523 + 2.95i)9-s + (1.07 − 0.358i)10-s − 2.34·11-s + (3.45 + 0.189i)12-s + 5.26·13-s + (−0.893 + 1.06i)15-s + (1.12 − 3.83i)16-s − 1.18i·17-s + (−3.72 + 2.02i)18-s − 7.12i·19-s + ⋯
L(s)  = 1  + (0.315 + 0.948i)2-s + (−0.766 − 0.642i)3-s + (−0.800 + 0.599i)4-s − 0.359i·5-s + (0.367 − 0.930i)6-s + (−0.821 − 0.569i)8-s + (0.174 + 0.984i)9-s + (0.340 − 0.113i)10-s − 0.707·11-s + (0.998 + 0.0547i)12-s + 1.46·13-s + (−0.230 + 0.275i)15-s + (0.281 − 0.959i)16-s − 0.287i·17-s + (−0.879 + 0.476i)18-s − 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 + 0.0547i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.998 + 0.0547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15230 - 0.0315413i\)
\(L(\frac12)\) \(\approx\) \(1.15230 - 0.0315413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.446 - 1.34i)T \)
3 \( 1 + (1.32 + 1.11i)T \)
7 \( 1 \)
good5 \( 1 + 0.803iT - 5T^{2} \)
11 \( 1 + 2.34T + 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 + 1.18iT - 17T^{2} \)
19 \( 1 + 7.12iT - 19T^{2} \)
23 \( 1 - 7.88T + 23T^{2} \)
29 \( 1 + 4.23iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 1.04T + 37T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 - 7.94iT - 43T^{2} \)
47 \( 1 + 6.09T + 47T^{2} \)
53 \( 1 + 8.72iT - 53T^{2} \)
59 \( 1 + 0.662T + 59T^{2} \)
61 \( 1 - 0.958T + 61T^{2} \)
67 \( 1 + 8.42iT - 67T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + 16.3iT - 89T^{2} \)
97 \( 1 + 4.37T + 97T^{2} \)
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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

−11.04397459639656231711413962088, −9.603406983085260125225724394421, −8.629603565724761381773538574368, −7.893593993890682937306174450634, −6.89282329662253449901573288680, −6.26035133915951681315631619437, −5.20299547935448570402140308101, −4.59410065390423805430276284778, −2.94175952503214804468344791840, −0.78347227996802227378881050278, 1.27295861066982493299156417910, 3.13551205661835194280870846020, 3.88403758750064425494718174365, 5.10325507741140256050094446649, 5.78791793991457463413701508141, 6.82550997070366373903524209071, 8.441049394588522196356855305936, 9.156352540532201046480336362544, 10.41693297020776204787083691312, 10.60128364881388564050739062822

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