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Properties

Label 2-585-13.3-c1-0-19
Degree $2$
Conductor $585$
Sign $-0.401 + 0.915i$
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  + (−0.313 − 0.542i)2-s + (0.803 − 1.39i)4-s + 5-s + (−2.21 + 3.83i)7-s − 2.26·8-s + (−0.313 − 0.542i)10-s + (−3.02 − 5.24i)11-s + (2.27 − 2.80i)13-s + 2.77·14-s + (−0.898 − 1.55i)16-s + (2.92 − 5.05i)17-s + (1.80 − 3.12i)19-s + (0.803 − 1.39i)20-s + (−1.89 + 3.28i)22-s + (−1.13 − 1.95i)23-s + ⋯
L(s)  = 1  + (−0.221 − 0.383i)2-s + (0.401 − 0.695i)4-s + 0.447·5-s + (−0.837 + 1.45i)7-s − 0.799·8-s + (−0.0991 − 0.171i)10-s + (−0.913 − 1.58i)11-s + (0.629 − 0.776i)13-s + 0.742·14-s + (−0.224 − 0.389i)16-s + (0.708 − 1.22i)17-s + (0.413 − 0.716i)19-s + (0.179 − 0.311i)20-s + (−0.404 + 0.701i)22-s + (−0.235 − 0.408i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.401 + 0.915i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646413 - 0.989093i\)
\(L(\frac12)\) \(\approx\) \(0.646413 - 0.989093i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (-2.27 + 2.80i)T \)
good2 \( 1 + (0.313 + 0.542i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.21 - 3.83i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.02 + 5.24i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.92 + 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.13 + 1.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.04 + 7.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.45T + 31T^{2} \)
37 \( 1 + (-0.898 - 1.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.24 - 5.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.22T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 + 2.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.16 - 3.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.24 - 7.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.819T + 73T^{2} \)
79 \( 1 + 8.42T + 79T^{2} \)
83 \( 1 - 2.35T + 83T^{2} \)
89 \( 1 + (0.386 + 0.669i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 + 5.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29320294575057747898372949032, −9.667747676795199349613147421695, −8.888005058639855975631408821325, −7.998141459004318348391460688191, −6.43595641229417738048622737914, −5.80445896143313874752251422607, −5.29528570803006147693409112089, −2.91354768050988957628691420427, −2.73814304254478478460589027174, −0.69843594285636750727066279333, 1.81544647822290738232505139816, 3.38912327447566268442115279191, 4.18099297643239660347377868047, 5.71821933856602873730027077643, 6.78314005838590830084075938010, 7.30081992354382502589821205358, 8.095528496786361692812514600842, 9.294588216893771859651578108735, 10.17846407797431878963646824252, 10.63748168655951209722069251234

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