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Properties

Label 2-585-65.9-c1-0-27
Degree $2$
Conductor $585$
Sign $0.583 - 0.812i$
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  + (2.20 + 1.27i)2-s + (2.24 + 3.88i)4-s + (0.817 − 2.08i)5-s + (2.54 − 1.46i)7-s + 6.31i·8-s + (4.45 − 3.54i)10-s + (−0.317 + 0.550i)11-s + (−3.60 + 0.0716i)13-s + 7.48·14-s + (−3.55 + 6.16i)16-s + (−1.05 + 0.611i)17-s + (0.682 + 1.18i)19-s + (9.90 − 1.49i)20-s + (−1.40 + 0.808i)22-s + (1.86 + 1.07i)23-s + ⋯
L(s)  = 1  + (1.55 + 0.900i)2-s + (1.12 + 1.94i)4-s + (0.365 − 0.930i)5-s + (0.961 − 0.555i)7-s + 2.23i·8-s + (1.40 − 1.12i)10-s + (−0.0957 + 0.165i)11-s + (−0.999 + 0.0198i)13-s + 1.99·14-s + (−0.889 + 1.54i)16-s + (−0.257 + 0.148i)17-s + (0.156 + 0.271i)19-s + (2.21 − 0.333i)20-s + (−0.298 + 0.172i)22-s + (0.388 + 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.583 - 0.812i$
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: \((2,\ 585,\ (\ :1/2),\ 0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.32414 + 1.70567i\)
\(L(\frac12)\) \(\approx\) \(3.32414 + 1.70567i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.817 + 2.08i)T \)
13 \( 1 + (3.60 - 0.0716i)T \)
good2 \( 1 + (-2.20 - 1.27i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-2.54 + 1.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.317 - 0.550i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.05 - 0.611i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.682 - 1.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (-1.05 - 0.611i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.98 - 8.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.18 - 0.683i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.95 - 4.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.31 - 2.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-6.27 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.8 + 7.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.23936937267309229284027675662, −9.969311241589587878180290150360, −8.768453420686136016038913384108, −7.78282590411210277501276176922, −7.19904090027332064862617043366, −6.02789926144465441460872180826, −4.98884289257095656214040294783, −4.74465475489091399522731898811, −3.54124645348894457645338672372, −1.91344589134087193778022050199, 1.94489639391473782580204083805, 2.64703013165463467774925814111, 3.79219265557474077910675325137, 5.00487004050443484109077756989, 5.54884799080309197609451219613, 6.64623514150139749877295485980, 7.60962090019198045781315875510, 9.122857886288509935315515536987, 10.14257008603387615127689281875, 10.96418681772355058661229671720

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