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Properties

Label 2-592-37.26-c1-0-14
Degree $2$
Conductor $592$
Sign $-0.284 + 0.958i$
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  + (1.45 − 2.51i)3-s + (−0.951 + 1.64i)5-s + (0.762 − 1.32i)7-s + (−2.71 − 4.70i)9-s + (1 − 1.73i)13-s + (2.76 + 4.78i)15-s + (−3.02 − 5.24i)17-s + (3.52 − 6.10i)19-s + (−2.21 − 3.83i)21-s − 4.28·23-s + (0.688 + 1.19i)25-s − 7.05·27-s − 1.28·29-s + 9.80·31-s + (1.45 + 2.51i)35-s + ⋯
L(s)  = 1  + (0.838 − 1.45i)3-s + (−0.425 + 0.737i)5-s + (0.288 − 0.499i)7-s + (−0.904 − 1.56i)9-s + (0.277 − 0.480i)13-s + (0.713 + 1.23i)15-s + (−0.733 − 1.27i)17-s + (0.808 − 1.40i)19-s + (−0.483 − 0.836i)21-s − 0.892·23-s + (0.137 + 0.238i)25-s − 1.35·27-s − 0.237·29-s + 1.76·31-s + (0.245 + 0.424i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.284 + 0.958i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03689 - 1.38874i\)
\(L(\frac12)\) \(\approx\) \(1.03689 - 1.38874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.854 - 6.02i)T \)
good3 \( 1 + (-1.45 + 2.51i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.951 - 1.64i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.762 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.02 + 5.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.52 + 6.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 - 9.80T + 31T^{2} \)
41 \( 1 + (0.811 - 1.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.76T + 43T^{2} \)
47 \( 1 - 5.80T + 47T^{2} \)
53 \( 1 + (-3.21 - 5.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.52 + 6.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.64 - 9.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.80 - 6.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.38 - 2.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + (-7.95 + 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.78 - 11.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.46 + 7.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−10.52859489345909318907554408837, −9.340029625701949953946377454496, −8.424995645199645803371389853779, −7.57371844148833973006823356312, −7.10120899095526490747838216475, −6.32413050468166625005520010282, −4.74056263244184752136693407596, −3.22879646531658436429469972922, −2.52097378135932899881242407030, −0.928286733360327462228718079348, 2.03411089462553261919438170985, 3.57082530832033825071633046048, 4.21169895280563722760617670185, 5.10221515831741566870480981949, 6.20244619252763561242306563166, 7.983041650235821952266297969513, 8.418374589741346458216437668526, 9.120141964607265122605776126664, 10.00499722994507504457572753033, 10.67013089740135467062602198146

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