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Properties

Label 2-592-148.119-c1-0-16
Degree $2$
Conductor $592$
Sign $0.283 + 0.958i$
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  + (1.28 − 2.22i)3-s + (3.31 − 0.887i)5-s + (0.921 + 0.532i)7-s + (−1.79 − 3.10i)9-s − 4.10·11-s + (4.46 − 1.19i)13-s + (2.27 − 8.50i)15-s + (−0.143 + 0.534i)17-s + (1.22 + 4.57i)19-s + (2.36 − 1.36i)21-s + (−3.27 + 3.27i)23-s + (5.85 − 3.38i)25-s − 1.49·27-s + (−6.59 + 6.59i)29-s + (−5.20 − 5.20i)31-s + ⋯
L(s)  = 1  + (0.740 − 1.28i)3-s + (1.48 − 0.397i)5-s + (0.348 + 0.201i)7-s + (−0.597 − 1.03i)9-s − 1.23·11-s + (1.23 − 0.331i)13-s + (0.588 − 2.19i)15-s + (−0.0347 + 0.129i)17-s + (0.281 + 1.04i)19-s + (0.516 − 0.297i)21-s + (−0.683 + 0.683i)23-s + (1.17 − 0.676i)25-s − 0.287·27-s + (−1.22 + 1.22i)29-s + (−0.935 − 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.283 + 0.958i$
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: \((2,\ 592,\ (\ :1/2),\ 0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88004 - 1.40439i\)
\(L(\frac12)\) \(\approx\) \(1.88004 - 1.40439i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.97 - 3.50i)T \)
good3 \( 1 + (-1.28 + 2.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.31 + 0.887i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.921 - 0.532i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.10T + 11T^{2} \)
13 \( 1 + (-4.46 + 1.19i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.143 - 0.534i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.22 - 4.57i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.27 - 3.27i)T - 23iT^{2} \)
29 \( 1 + (6.59 - 6.59i)T - 29iT^{2} \)
31 \( 1 + (5.20 + 5.20i)T + 31iT^{2} \)
41 \( 1 + (-4.62 - 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.34 - 1.34i)T - 43iT^{2} \)
47 \( 1 + 9.47iT - 47T^{2} \)
53 \( 1 + (-1.36 - 2.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.84 - 1.29i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.42 + 9.04i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.43 + 4.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.19 - 1.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + (-0.448 - 1.67i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (12.9 - 7.50i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (14.7 + 3.95i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.15 - 9.15i)T + 97iT^{2} \)
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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

−10.37850148895844503688173197302, −9.523949217845094375421423144100, −8.516602455338739184497028188826, −8.020844923360708126611290143264, −7.00487575857709625632833147284, −5.80485317372474589901115106056, −5.43434381142181194141541472523, −3.43815170792118591810539181461, −2.10937856728691549206558388705, −1.50650737538668986674625727323, 2.06657784413001024862073648992, 3.04899259874877309286990954199, 4.23993309037218672559268335407, 5.28569256047253570685479911480, 6.08894735384392448435744817052, 7.40260031505586536289802577663, 8.593864682319833590744720314557, 9.221531209029741143897203597655, 9.984356619663978571249609197221, 10.66765597816748579447211777127

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