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Properties

Label 2-592-592.51-c1-0-18
Degree $2$
Conductor $592$
Sign $0.859 - 0.511i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + (−1.40 − 0.162i)2-s + (0.663 − 2.47i)3-s + (1.94 + 0.455i)4-s + (−1.07 + 0.623i)5-s + (−1.33 + 3.37i)6-s + (2.26 + 3.92i)7-s + (−2.66 − 0.956i)8-s + (−3.09 − 1.78i)9-s + (1.61 − 0.700i)10-s + (−0.381 + 0.381i)11-s + (2.42 − 4.52i)12-s + (2.15 + 3.73i)13-s + (−2.54 − 5.87i)14-s + (0.827 + 3.08i)15-s + (3.58 + 1.77i)16-s + (0.915 + 3.41i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.114i)2-s + (0.383 − 1.43i)3-s + (0.973 + 0.227i)4-s + (−0.482 + 0.278i)5-s + (−0.544 + 1.37i)6-s + (0.855 + 1.48i)7-s + (−0.941 − 0.338i)8-s + (−1.03 − 0.596i)9-s + (0.511 − 0.221i)10-s + (−0.114 + 0.114i)11-s + (0.699 − 1.30i)12-s + (0.598 + 1.03i)13-s + (−0.680 − 1.57i)14-s + (0.213 + 0.797i)15-s + (0.896 + 0.443i)16-s + (0.222 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880635 + 0.242193i\)
\(L(\frac12)\) \(\approx\) \(0.880635 + 0.242193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.162i)T \)
37 \( 1 + (-4.23 + 4.36i)T \)
good3 \( 1 + (-0.663 + 2.47i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.07 - 0.623i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.26 - 3.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.381 - 0.381i)T - 11iT^{2} \)
13 \( 1 + (-2.15 - 3.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.915 - 3.41i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.77 - 3.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.33 - 5.33i)T - 23iT^{2} \)
29 \( 1 - 4.35iT - 29T^{2} \)
31 \( 1 + (-3.12 + 3.12i)T - 31iT^{2} \)
41 \( 1 + (0.975 + 1.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.98T + 43T^{2} \)
47 \( 1 + 5.57iT - 47T^{2} \)
53 \( 1 + (1.51 + 5.66i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.04 + 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.36 + 1.94i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.81 + 10.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.700 - 1.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.94T + 73T^{2} \)
79 \( 1 + (1.78 - 6.66i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.944 - 3.52i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-3.15 - 11.7i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.0699 - 0.0699i)T - 97iT^{2} \)
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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

−11.01031659655062995499061133471, −9.657227183950536340643139751319, −8.531752854731752263444522513852, −8.289560271694021451111784055814, −7.50635348081696607118949555149, −6.45216762679553415309923046671, −5.79910318390201353304657910952, −3.72947395165768948834172559556, −2.08249986830820359073145540237, −1.80665927598845103212048163348, 0.67134121062483540652028663662, 2.74984308182745549955480570939, 4.11622393383588196415609134380, 4.66620835569559117875270411661, 6.16384587883785646153132348000, 7.44963235030453727193630751083, 8.261591695924346313496761129456, 8.696133448885123682095436632788, 10.01087682598814018815668155791, 10.38054109706930797987905545245

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