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Properties

Label 2-597-597.98-c0-0-0
Degree $2$
Conductor $597$
Sign $0.530 + 0.847i$
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  + (−0.995 + 0.0950i)3-s + (0.235 − 0.971i)4-s + (−0.379 + 0.532i)7-s + (0.981 − 0.189i)9-s + (−0.142 + 0.989i)12-s + (1.56 − 1.23i)13-s + (−0.888 − 0.458i)16-s + (−0.928 − 1.60i)19-s + (0.327 − 0.566i)21-s + (0.415 − 0.909i)25-s + (−0.959 + 0.281i)27-s + (0.428 + 0.494i)28-s + (0.995 + 0.0950i)31-s + (0.0475 − 0.998i)36-s + (−0.841 + 1.45i)37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)3-s + (0.235 − 0.971i)4-s + (−0.379 + 0.532i)7-s + (0.981 − 0.189i)9-s + (−0.142 + 0.989i)12-s + (1.56 − 1.23i)13-s + (−0.888 − 0.458i)16-s + (−0.928 − 1.60i)19-s + (0.327 − 0.566i)21-s + (0.415 − 0.909i)25-s + (−0.959 + 0.281i)27-s + (0.428 + 0.494i)28-s + (0.995 + 0.0950i)31-s + (0.0475 − 0.998i)36-s + (−0.841 + 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7031511712\)
\(L(\frac12)\) \(\approx\) \(0.7031511712\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
199 \( 1 + (0.142 - 0.989i)T \)
good2 \( 1 + (-0.235 + 0.971i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.379 - 0.532i)T + (-0.327 - 0.945i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-1.56 + 1.23i)T + (0.235 - 0.971i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.888 - 0.458i)T^{2} \)
29 \( 1 + (-0.723 + 0.690i)T^{2} \)
31 \( 1 + (-0.995 - 0.0950i)T + (0.981 + 0.189i)T^{2} \)
37 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.928 + 0.371i)T^{2} \)
43 \( 1 + (-0.959 - 1.66i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.327 + 0.945i)T^{2} \)
53 \( 1 + (-0.235 - 0.971i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (1.49 + 0.961i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.975 + 0.627i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.995 + 0.0950i)T^{2} \)
73 \( 1 + (-0.462 - 1.90i)T + (-0.888 + 0.458i)T^{2} \)
79 \( 1 + (-0.252 - 0.130i)T + (0.580 + 0.814i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.723 + 0.690i)T^{2} \)
97 \( 1 + (1.44 + 0.137i)T + (0.981 + 0.189i)T^{2} \)
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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

−10.84528891500366138008153321270, −10.11956068684037612045562514155, −9.179320904990316541087711693305, −8.178151470406046543107017127113, −6.57058775441873797800212445878, −6.32885853657602086311782919063, −5.35796986669086532407028646126, −4.47114437961837182664620633408, −2.80328562907065634704654265181, −1.03329902326487641630505538420, 1.76217896991753915268548477056, 3.71523006109055603876287680190, 4.18427904798751955554706856234, 5.77116772227918200305597092394, 6.59112525880662106645001611252, 7.25765284553614721128692076596, 8.337559298128547567916297494516, 9.227633962098188129303111025821, 10.54738420083469991848091136807, 10.95720397005824353698276616677

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