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Properties

Label 2-585-117.25-c1-0-38
Degree $2$
Conductor $585$
Sign $-0.507 - 0.861i$
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  + (−1.64 − 0.947i)2-s + (−1.39 − 1.02i)3-s + (0.795 + 1.37i)4-s + (0.866 − 0.5i)5-s + (1.32 + 3.00i)6-s + (2.82 + 1.63i)7-s + 0.773i·8-s + (0.904 + 2.86i)9-s − 1.89·10-s + (−0.866 − 0.500i)11-s + (0.298 − 2.74i)12-s + (−2.87 − 2.17i)13-s + (−3.09 − 5.35i)14-s + (−1.72 − 0.187i)15-s + (2.32 − 4.02i)16-s − 7.42·17-s + ⋯
L(s)  = 1  + (−1.16 − 0.670i)2-s + (−0.806 − 0.590i)3-s + (0.397 + 0.689i)4-s + (0.387 − 0.223i)5-s + (0.540 + 1.22i)6-s + (1.06 + 0.616i)7-s + 0.273i·8-s + (0.301 + 0.953i)9-s − 0.599·10-s + (−0.261 − 0.150i)11-s + (0.0863 − 0.791i)12-s + (−0.797 − 0.602i)13-s + (−0.826 − 1.43i)14-s + (−0.444 − 0.0484i)15-s + (0.581 − 1.00i)16-s − 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0321654 + 0.0562607i\)
\(L(\frac12)\) \(\approx\) \(0.0321654 + 0.0562607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 + 1.02i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 + (1.64 + 0.947i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-2.82 - 1.63i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.866 + 0.500i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 7.42T + 17T^{2} \)
19 \( 1 - 2.59iT - 19T^{2} \)
23 \( 1 + (4.12 + 7.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.59 - 4.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.82 - 2.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.25iT - 37T^{2} \)
41 \( 1 + (-3.27 + 1.88i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.95 - 8.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.84 + 5.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.76T + 53T^{2} \)
59 \( 1 + (5.16 - 2.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.45 + 5.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.37 - 1.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 - 4.10iT - 73T^{2} \)
79 \( 1 + (-1.42 + 2.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.63 - 1.52i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (-4.09 - 2.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38425263507204159310927922711, −9.246505961299013873628559725439, −8.390714270071201454799628519430, −7.81907286627386432667316437674, −6.58791841933377791374114588285, −5.41993533941938475255534944253, −4.77142912632755930959975677587, −2.38269765654409013082160898458, −1.72806503618506497295211695619, −0.05693910526281265577815347186, 1.80498553762035385749357045633, 4.03094404785495615331820444349, 4.88282749578433195162098285179, 6.06855538415758389573832156881, 7.01089428434873869566991486302, 7.61843395008040779142858899763, 8.789986525762114010958353646981, 9.571838120274375099550563544194, 10.16330463274465779099924234697, 11.17472066042038583857642794496

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