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Properties

Label 2-597-597.251-c0-0-0
Degree $2$
Conductor $597$
Sign $0.983 + 0.180i$
Analytic cond. $0.297941$
Root an. cond. $0.545840$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.981 − 0.189i)3-s + (−0.888 − 0.458i)4-s + (0.514 + 1.48i)7-s + (0.928 − 0.371i)9-s + (−0.959 − 0.281i)12-s + (0.462 − 1.90i)13-s + (0.580 + 0.814i)16-s + (−0.723 + 1.25i)19-s + (0.786 + 1.36i)21-s + (−0.654 − 0.755i)25-s + (0.841 − 0.540i)27-s + (0.223 − 1.55i)28-s + (−0.981 − 0.189i)31-s + (−0.995 − 0.0950i)36-s + (−0.415 − 0.719i)37-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)3-s + (−0.888 − 0.458i)4-s + (0.514 + 1.48i)7-s + (0.928 − 0.371i)9-s + (−0.959 − 0.281i)12-s + (0.462 − 1.90i)13-s + (0.580 + 0.814i)16-s + (−0.723 + 1.25i)19-s + (0.786 + 1.36i)21-s + (−0.654 − 0.755i)25-s + (0.841 − 0.540i)27-s + (0.223 − 1.55i)28-s + (−0.981 − 0.189i)31-s + (−0.995 − 0.0950i)36-s + (−0.415 − 0.719i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.983 + 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094602574\)
\(L(\frac12)\) \(\approx\) \(1.094602574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 + 0.189i)T \)
199 \( 1 + (0.959 + 0.281i)T \)
good2 \( 1 + (0.888 + 0.458i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.514 - 1.48i)T + (-0.786 + 0.618i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.462 + 1.90i)T + (-0.888 - 0.458i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (-0.0475 + 0.998i)T^{2} \)
31 \( 1 + (0.981 + 0.189i)T + (0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.723 + 0.690i)T^{2} \)
43 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.786 - 0.618i)T^{2} \)
53 \( 1 + (0.888 - 0.458i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.271 - 0.595i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (1.65 - 0.850i)T + (0.580 - 0.814i)T^{2} \)
79 \( 1 + (1.11 + 1.56i)T + (-0.327 + 0.945i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (-0.0475 + 0.998i)T^{2} \)
97 \( 1 + (-0.0934 - 0.0180i)T + (0.928 + 0.371i)T^{2} \)
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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

−10.53192536424055037252312608265, −9.906806999863819697853474973094, −8.882045404568197415433946097905, −8.375192993732060963248949204336, −7.78666346281478284328134099328, −6.01795001205216703670274061275, −5.47979836970515097540742580683, −4.19428327037249191171854278057, −3.03651457358576234851189256776, −1.73506629156676794630808999884, 1.75110852212917703615688212007, 3.53081548541162962158160229943, 4.18980085326544501759826317292, 4.85978813163436799137370621194, 6.86816369649743725840553678141, 7.39437795821090062717789289921, 8.472628777829646440943286970743, 9.031921669745699683528598606642, 9.833918619897525827255362681681, 10.80582346275724158292998809289

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