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Properties

Label 2-588-12.11-c1-0-67
Degree $2$
Conductor $588$
Sign $-0.755 - 0.655i$
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  + (−0.750 − 1.19i)2-s + (0.448 − 1.67i)3-s + (−0.874 + 1.79i)4-s − 3.56i·5-s + (−2.34 + 0.717i)6-s + (2.81 − 0.301i)8-s + (−2.59 − 1.50i)9-s + (−4.27 + 2.67i)10-s + 0.335·11-s + (2.61 + 2.26i)12-s − 3.34·13-s + (−5.96 − 1.59i)15-s + (−2.47 − 3.14i)16-s − 0.335i·17-s + (0.150 + 4.23i)18-s − 1.84i·19-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)2-s + (0.258 − 0.965i)3-s + (−0.437 + 0.899i)4-s − 1.59i·5-s + (−0.956 + 0.292i)6-s + (0.994 − 0.106i)8-s + (−0.865 − 0.500i)9-s + (−1.35 + 0.845i)10-s + 0.101·11-s + (0.755 + 0.655i)12-s − 0.928·13-s + (−1.53 − 0.412i)15-s + (−0.617 − 0.786i)16-s − 0.0813i·17-s + (0.0354 + 0.999i)18-s − 0.424i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.755 - 0.655i$
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: \((2,\ 588,\ (\ :1/2),\ -0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299249 + 0.802027i\)
\(L(\frac12)\) \(\approx\) \(0.299249 + 0.802027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.750 + 1.19i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
7 \( 1 \)
good5 \( 1 + 3.56iT - 5T^{2} \)
11 \( 1 - 0.335T + 11T^{2} \)
13 \( 1 + 3.34T + 13T^{2} \)
17 \( 1 + 0.335iT - 17T^{2} \)
19 \( 1 + 1.84iT - 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + 5.91iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + 8.91T + 47T^{2} \)
53 \( 1 + 4.79iT - 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 - 0.353T + 61T^{2} \)
67 \( 1 + 3.19iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 4.69T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 6.89T + 83T^{2} \)
89 \( 1 - 3.87iT - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.900287933909549962634619808644, −9.206596361638590755009870605791, −8.501598097836834293967461274165, −7.83325277977994722067133664905, −6.84601469135721764036948652590, −5.30947307153381941027763597087, −4.41339031241194199826627372673, −2.92199696086430359357669246936, −1.69200323861738231278646071240, −0.55805113252775934264538799508, 2.44809678025209432961904113088, 3.64686009553084169379046143680, 4.88639988946355243790710560186, 5.89128940257090269991457176447, 6.90275085219696646556011821385, 7.58731289075767848906451831079, 8.629203910919776258815569269395, 9.606368907711503880337289558729, 10.18683752021523188722057406802, 10.84629116150756918680547935406

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