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Properties

Label 2-592-148.119-c1-0-14
Degree $2$
Conductor $592$
Sign $0.691 + 0.722i$
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  + (1.06 − 1.84i)3-s + (1.91 − 0.512i)5-s + (1.95 + 1.12i)7-s + (−0.779 − 1.35i)9-s + 5.67·11-s + (−3.75 + 1.00i)13-s + (1.09 − 4.08i)15-s + (−0.686 + 2.56i)17-s + (0.0131 + 0.0491i)19-s + (4.16 − 2.40i)21-s + (−1.92 + 1.92i)23-s + (−0.932 + 0.538i)25-s + 3.07·27-s + (−3.06 + 3.06i)29-s + (−4.81 − 4.81i)31-s + ⋯
L(s)  = 1  + (0.616 − 1.06i)3-s + (0.855 − 0.229i)5-s + (0.737 + 0.425i)7-s + (−0.259 − 0.450i)9-s + 1.71·11-s + (−1.04 + 0.279i)13-s + (0.282 − 1.05i)15-s + (−0.166 + 0.621i)17-s + (0.00302 + 0.0112i)19-s + (0.908 − 0.524i)21-s + (−0.401 + 0.401i)23-s + (−0.186 + 0.107i)25-s + 0.591·27-s + (−0.568 + 0.568i)29-s + (−0.865 − 0.865i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08001 - 0.887629i\)
\(L(\frac12)\) \(\approx\) \(2.08001 - 0.887629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (3.26 + 5.13i)T \)
good3 \( 1 + (-1.06 + 1.84i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.91 + 0.512i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.95 - 1.12i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.67T + 11T^{2} \)
13 \( 1 + (3.75 - 1.00i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.686 - 2.56i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.0131 - 0.0491i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.92 - 1.92i)T - 23iT^{2} \)
29 \( 1 + (3.06 - 3.06i)T - 29iT^{2} \)
31 \( 1 + (4.81 + 4.81i)T + 31iT^{2} \)
41 \( 1 + (7.86 + 4.53i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 + 2.99i)T - 43iT^{2} \)
47 \( 1 - 0.988iT - 47T^{2} \)
53 \( 1 + (1.98 + 3.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.88 + 2.11i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.60 - 9.72i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.22 - 2.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.1 + 5.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + (-2.08 - 7.76i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-8.83 + 5.10i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.54 - 2.02i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.3 - 12.3i)T + 97iT^{2} \)
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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

−10.53752787791258831989552495101, −9.290211469944048988163029377159, −8.978207616037308349128600742388, −7.85061266873893995102475202694, −7.07691053817890423801665189177, −6.16036147532792220558070888642, −5.14413031454317591876600153909, −3.78969015900775721353647161286, −2.04810252404442588169913998647, −1.67436475884226875524566824940, 1.72026073835643495702305678034, 3.12135464618200276004426110000, 4.23186813982570943629301276730, 4.96623874923978951495298289948, 6.27366560367247102248442931951, 7.22356374534953883249761136764, 8.403619110348439222979949142168, 9.338013829771184352437877435258, 9.754913879765169073222078893807, 10.54091720876043542062383954644

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