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Properties

Label 2-588-12.11-c1-0-19
Degree $2$
Conductor $588$
Sign $-0.992 - 0.122i$
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  + (1.05 + 0.941i)2-s + (−0.406 + 1.68i)3-s + (0.227 + 1.98i)4-s + 1.25i·5-s + (−2.01 + 1.39i)6-s + (−1.63 + 2.31i)8-s + (−2.66 − 1.36i)9-s + (−1.18 + 1.32i)10-s + 5.05·11-s + (−3.43 − 0.425i)12-s − 4.41·13-s + (−2.11 − 0.509i)15-s + (−3.89 + 0.903i)16-s + 5.85i·17-s + (−1.52 − 3.95i)18-s − 1.53i·19-s + ⋯
L(s)  = 1  + (0.746 + 0.665i)2-s + (−0.234 + 0.972i)3-s + (0.113 + 0.993i)4-s + 0.560i·5-s + (−0.822 + 0.569i)6-s + (−0.576 + 0.817i)8-s + (−0.889 − 0.456i)9-s + (−0.373 + 0.418i)10-s + 1.52·11-s + (−0.992 − 0.122i)12-s − 1.22·13-s + (−0.544 − 0.131i)15-s + (−0.974 + 0.225i)16-s + 1.41i·17-s + (−0.359 − 0.932i)18-s − 0.351i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.992 - 0.122i$
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: \((2,\ 588,\ (\ :1/2),\ -0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112904 + 1.83219i\)
\(L(\frac12)\) \(\approx\) \(0.112904 + 1.83219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 0.941i)T \)
3 \( 1 + (0.406 - 1.68i)T \)
7 \( 1 \)
good5 \( 1 - 1.25iT - 5T^{2} \)
11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 - 5.85iT - 17T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
23 \( 1 - 0.313T + 23T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + 5.33T + 37T^{2} \)
41 \( 1 - 1.97iT - 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 - 7.96T + 59T^{2} \)
61 \( 1 - 0.302T + 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 - 4.53T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 + 6.78T + 83T^{2} \)
89 \( 1 + 4.79iT - 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−11.24772717929784067442437237342, −10.27096207271558459460520399866, −9.309978065515481862340167049244, −8.511868244539698228461412908832, −7.29600602468781049711217767216, −6.42319941955765298800813928748, −5.67266526568764498848257982227, −4.44168374298499698516753320148, −3.86189260152901151397196899033, −2.63391546142935928010236502115, 0.859615978217787340632515720423, 2.07650750288834658034509833195, 3.36102873136614099410116820189, 4.78888133134056613600087184303, 5.42311684356944404787281227564, 6.73278658556722505486948320768, 7.16365299552517241951998780102, 8.746120577254554588031289197704, 9.380362316569994107419674540183, 10.50903691271194938238936985135

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