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Properties

Label 2-585-65.9-c1-0-25
Degree $2$
Conductor $585$
Sign $-0.651 + 0.758i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + (−0.521 − 0.301i)2-s + (−0.818 − 1.41i)4-s + (−0.446 − 2.19i)5-s + (3.08 − 1.77i)7-s + 2.18i·8-s + (−0.426 + 1.27i)10-s + (1.30 − 2.26i)11-s + (2.88 + 2.16i)13-s − 2.14·14-s + (−0.978 + 1.69i)16-s + (1.94 − 1.12i)17-s + (−4.00 − 6.94i)19-s + (−2.74 + 2.42i)20-s + (−1.36 + 0.787i)22-s + (1.43 + 0.826i)23-s + ⋯
L(s)  = 1  + (−0.368 − 0.212i)2-s + (−0.409 − 0.709i)4-s + (−0.199 − 0.979i)5-s + (1.16 − 0.672i)7-s + 0.774i·8-s + (−0.134 + 0.403i)10-s + (0.394 − 0.682i)11-s + (0.800 + 0.599i)13-s − 0.572·14-s + (−0.244 + 0.423i)16-s + (0.471 − 0.272i)17-s + (−0.919 − 1.59i)19-s + (−0.613 + 0.542i)20-s + (−0.290 + 0.167i)22-s + (0.298 + 0.172i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464504 - 1.01171i\)
\(L(\frac12)\) \(\approx\) \(0.464504 - 1.01171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.446 + 2.19i)T \)
13 \( 1 + (-2.88 - 2.16i)T \)
good2 \( 1 + (0.521 + 0.301i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.08 + 1.77i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.30 + 2.26i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.00 + 6.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.43 - 0.826i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.26 - 3.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 + (3.64 + 2.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.388 - 0.673i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.197 - 0.113i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.14iT - 47T^{2} \)
53 \( 1 + 6.42iT - 53T^{2} \)
59 \( 1 + (-6.26 - 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.969 - 0.559i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.66 + 9.81i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (-8.60 + 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 6.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.55609417640748738460653403322, −9.192901567812212633579008353464, −8.850875691669945333905100047010, −8.047434342569665101607065159380, −6.83623494673622995014730153831, −5.51740417637263130033505745383, −4.79787395842098338942078815850, −3.90854483982708616612951786736, −1.76011024648474307094800963757, −0.77672431358026185321527625655, 1.90638026889014882530732496887, 3.40668632552929546086526638447, 4.26346550541674784935272120463, 5.63702789527150342790608225935, 6.65887561234024816084236064802, 7.83924461728469854785525280773, 8.103706819015108312519177163141, 9.112830847355752444777845996214, 10.17626258527225363178498942261, 10.93710680783392306932251747349

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