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Properties

Label 2-588-196.27-c1-0-18
Degree $2$
Conductor $588$
Sign $0.633 + 0.773i$
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  + (0.0367 − 1.41i)2-s + (−0.222 + 0.974i)3-s + (−1.99 − 0.103i)4-s + (−3.11 − 0.710i)5-s + (1.37 + 0.350i)6-s + (2.64 + 0.126i)7-s + (−0.220 + 2.81i)8-s + (−0.900 − 0.433i)9-s + (−1.11 + 4.37i)10-s + (2.20 + 4.58i)11-s + (0.545 − 1.92i)12-s + (−1.35 − 2.80i)13-s + (0.275 − 3.73i)14-s + (1.38 − 2.87i)15-s + (3.97 + 0.414i)16-s + (4.52 − 3.60i)17-s + ⋯
L(s)  = 1  + (0.0259 − 0.999i)2-s + (−0.128 + 0.562i)3-s + (−0.998 − 0.0519i)4-s + (−1.39 − 0.317i)5-s + (0.559 + 0.143i)6-s + (0.998 + 0.0477i)7-s + (−0.0778 + 0.996i)8-s + (−0.300 − 0.144i)9-s + (−0.353 + 1.38i)10-s + (0.665 + 1.38i)11-s + (0.157 − 0.555i)12-s + (−0.374 − 0.778i)13-s + (0.0736 − 0.997i)14-s + (0.357 − 0.742i)15-s + (0.994 + 0.103i)16-s + (1.09 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.633 + 0.773i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.633 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01907 - 0.482757i\)
\(L(\frac12)\) \(\approx\) \(1.01907 - 0.482757i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0367 + 1.41i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-2.64 - 0.126i)T \)
good5 \( 1 + (3.11 + 0.710i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-2.20 - 4.58i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (1.35 + 2.80i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-4.52 + 3.60i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + (5.32 + 4.24i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-2.36 - 2.96i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 5.86T + 31T^{2} \)
37 \( 1 + (0.909 + 1.14i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-2.89 - 0.660i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-9.85 + 2.25i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.73 + 1.80i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (6.48 - 8.12i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-1.87 - 8.23i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-3.00 + 2.39i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 - 4.00iT - 67T^{2} \)
71 \( 1 + (4.57 + 3.64i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-3.05 + 6.34i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (2.31 + 1.11i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-3.82 + 7.93i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 1.22iT - 97T^{2} \)
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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

−10.60558814273519116618735396493, −9.882852748757167002281824608343, −9.008658126781059253181996268493, −7.944345886712563951555994321302, −7.48093514744643909239878515712, −5.42343108769590944670015264960, −4.61549795949677370896891814673, −4.03338283928004387377770614670, −2.77629946065231789070621864440, −0.970385818101806175489410586203, 1.01916093055155127730300569547, 3.44062529422585547444019302972, 4.24796182383238757949621752529, 5.50759429593928933234714073798, 6.37324986855400541710423625360, 7.53183450125905288400307235548, 7.890321435673653195862822702725, 8.564130662959319332527029582248, 9.751059896511024953431212835859, 11.09640669622687784075470393149

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