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Properties

Label 2-585-9.7-c1-0-14
Degree $2$
Conductor $585$
Sign $0.206 - 0.978i$
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  + (−0.643 + 1.11i)2-s + (−1.73 − 0.0293i)3-s + (0.170 + 0.296i)4-s + (−0.5 − 0.866i)5-s + (1.14 − 1.91i)6-s + (0.391 − 0.678i)7-s − 3.01·8-s + (2.99 + 0.101i)9-s + 1.28·10-s + (0.972 − 1.68i)11-s + (−0.287 − 0.517i)12-s + (0.5 + 0.866i)13-s + (0.504 + 0.873i)14-s + (0.840 + 1.51i)15-s + (1.59 − 2.77i)16-s + 0.858·17-s + ⋯
L(s)  = 1  + (−0.455 + 0.788i)2-s + (−0.999 − 0.0169i)3-s + (0.0854 + 0.148i)4-s + (−0.223 − 0.387i)5-s + (0.468 − 0.780i)6-s + (0.148 − 0.256i)7-s − 1.06·8-s + (0.999 + 0.0338i)9-s + 0.407·10-s + (0.293 − 0.507i)11-s + (−0.0829 − 0.149i)12-s + (0.138 + 0.240i)13-s + (0.134 + 0.233i)14-s + (0.217 + 0.391i)15-s + (0.399 − 0.692i)16-s + 0.208·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646263 + 0.523881i\)
\(L(\frac12)\) \(\approx\) \(0.646263 + 0.523881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0293i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.643 - 1.11i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.391 + 0.678i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.972 + 1.68i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.858T + 17T^{2} \)
19 \( 1 - 5.16T + 19T^{2} \)
23 \( 1 + (-1.28 - 2.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.344 - 0.596i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.29 - 3.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 + (-1.47 - 2.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.65 - 6.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.12 - 3.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.78T + 53T^{2} \)
59 \( 1 + (2.70 + 4.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.29 - 7.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.687 - 1.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.64T + 71T^{2} \)
73 \( 1 - 8.86T + 73T^{2} \)
79 \( 1 + (-2.69 + 4.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.24 - 3.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-8.35 + 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11235345658831159153424038576, −9.838787735130392090435729821371, −9.094575053636922513288938065448, −7.987720762874334667572116314092, −7.35401055970106127632141748587, −6.41533292115670095282099427602, −5.65246374609065257917272390652, −4.56910844910947983670077335756, −3.27508982083024851863934074239, −1.04952788296707336990827791323, 0.826349412480870381318034773504, 2.24233044101560885244514679257, 3.64904357318831865517193406154, 5.01826852615967525404182458130, 5.94834995508929220998908013091, 6.80314149636648505457900296963, 7.81355017374881288149990867878, 9.160392847005561458670514450095, 9.919581466958783191448741189291, 10.55461114415608305383101580841

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