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Properties

Label 2-592-148.119-c1-0-0
Degree $2$
Conductor $592$
Sign $0.271 + 0.962i$
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  + (−1.69 + 2.92i)3-s + (−3.56 + 0.954i)5-s + (1.92 + 1.11i)7-s + (−4.21 − 7.30i)9-s − 2.24·11-s + (−1.64 + 0.440i)13-s + (3.22 − 12.0i)15-s + (−1.39 + 5.22i)17-s + (0.234 + 0.873i)19-s + (−6.51 + 3.76i)21-s + (2.13 − 2.13i)23-s + (7.44 − 4.30i)25-s + 18.3·27-s + (4.32 − 4.32i)29-s + (2.78 + 2.78i)31-s + ⋯
L(s)  = 1  + (−0.976 + 1.69i)3-s + (−1.59 + 0.426i)5-s + (0.728 + 0.420i)7-s + (−1.40 − 2.43i)9-s − 0.676·11-s + (−0.455 + 0.122i)13-s + (0.833 − 3.11i)15-s + (−0.339 + 1.26i)17-s + (0.0537 + 0.200i)19-s + (−1.42 + 0.821i)21-s + (0.446 − 0.446i)23-s + (1.48 − 0.860i)25-s + 3.53·27-s + (0.802 − 0.802i)29-s + (0.499 + 0.499i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.271 + 0.962i$
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: \((2,\ 592,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0354917 - 0.0268590i\)
\(L(\frac12)\) \(\approx\) \(0.0354917 - 0.0268590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.473 + 6.06i)T \)
good3 \( 1 + (1.69 - 2.92i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.56 - 0.954i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.92 - 1.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + (1.64 - 0.440i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.39 - 5.22i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.234 - 0.873i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.13 + 2.13i)T - 23iT^{2} \)
29 \( 1 + (-4.32 + 4.32i)T - 29iT^{2} \)
31 \( 1 + (-2.78 - 2.78i)T + 31iT^{2} \)
41 \( 1 + (10.1 + 5.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.347 + 0.347i)T - 43iT^{2} \)
47 \( 1 - 9.31iT - 47T^{2} \)
53 \( 1 + (-1.97 - 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.10 + 1.90i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.0255 - 0.0954i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.46 + 2.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.21 + 4.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.21iT - 73T^{2} \)
79 \( 1 + (-4.01 - 14.9i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (7.19 - 4.15i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.39 + 2.24i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.02 + 6.02i)T + 97iT^{2} \)
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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

−11.01900386998702299688562166156, −10.82834161585258527918019574907, −9.932622329307371512235777729479, −8.701318897833621427295097110025, −8.099806078114708122791848853867, −6.78427330083386759717649201145, −5.64711154997655444664632817325, −4.67202673068205187347794927885, −4.12943657357965526337625423273, −3.05564364075606758468092601794, 0.03232052745822163332106776662, 1.19019815640646619038240333261, 2.84907538255186268594370727056, 4.75418329627148825844882672476, 5.14929697743667666225192380044, 6.72441200734157120820744601330, 7.33662751913024255271966786461, 7.933644721971326745600536532671, 8.569528588129378261175431681135, 10.39952407678382301433636899272

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