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Properties

Label 2-585-9.4-c1-0-18
Degree $2$
Conductor $585$
Sign $0.308 + 0.951i$
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  + (−1.09 − 1.89i)2-s + (1.72 − 0.119i)3-s + (−1.39 + 2.40i)4-s + (0.5 − 0.866i)5-s + (−2.11 − 3.14i)6-s + (2.12 + 3.68i)7-s + 1.70·8-s + (2.97 − 0.414i)9-s − 2.18·10-s + (0.344 + 0.596i)11-s + (−2.11 + 4.32i)12-s + (−0.5 + 0.866i)13-s + (4.65 − 8.05i)14-s + (0.760 − 1.55i)15-s + (0.913 + 1.58i)16-s + 2.89·17-s + ⋯
L(s)  = 1  + (−0.773 − 1.33i)2-s + (0.997 − 0.0692i)3-s + (−0.695 + 1.20i)4-s + (0.223 − 0.387i)5-s + (−0.863 − 1.28i)6-s + (0.803 + 1.39i)7-s + 0.604·8-s + (0.990 − 0.138i)9-s − 0.691·10-s + (0.103 + 0.179i)11-s + (−0.610 + 1.24i)12-s + (−0.138 + 0.240i)13-s + (1.24 − 2.15i)14-s + (0.196 − 0.401i)15-s + (0.228 + 0.395i)16-s + 0.702·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25288 - 0.911245i\)
\(L(\frac12)\) \(\approx\) \(1.25288 - 0.911245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.119i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.09 + 1.89i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-2.12 - 3.68i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.344 - 0.596i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.89T + 17T^{2} \)
19 \( 1 - 1.63T + 19T^{2} \)
23 \( 1 + (-0.480 + 0.831i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 + 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.953 - 1.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 + (3.93 - 6.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.78 + 6.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.08 + 7.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + (1.44 - 2.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.50 + 4.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 - 4.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.84T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + (6.12 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.07 + 12.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + (-6.83 - 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.23586975214480897575961037034, −9.675354311760119537027811472855, −8.754508051233713914807228396308, −8.519115119163958841295774616089, −7.45195315678208147560172360248, −5.82122738602859427467591560469, −4.59041499102320543594077887943, −3.25243112273633220790050807031, −2.28193337934163942122796453188, −1.48575238841876389381488469533, 1.27268660042557720610888761672, 3.16070414652137962902967776591, 4.39002639529222226366241643854, 5.59384378431126853377769918895, 6.88145654555659669575816315445, 7.46964628777225983054606418640, 7.993952001529892921902739889366, 8.879259758603297870360025016054, 9.844992573640750808027417511617, 10.35344017886091287661236867271

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