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Properties

Label 2-585-65.57-c1-0-26
Degree $2$
Conductor $585$
Sign $0.990 - 0.136i$
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  + 2.56·2-s + 4.57·4-s + (1.43 + 1.71i)5-s − 1.73i·7-s + 6.61·8-s + (3.68 + 4.39i)10-s + (−1.61 − 1.61i)11-s + (−3.55 + 0.582i)13-s − 4.46i·14-s + 7.80·16-s + (−2.35 + 2.35i)17-s + (−4.94 − 4.94i)19-s + (6.57 + 7.84i)20-s + (−4.15 − 4.15i)22-s + (5.08 + 5.08i)23-s + ⋯
L(s)  = 1  + 1.81·2-s + 2.28·4-s + (0.642 + 0.766i)5-s − 0.657i·7-s + 2.33·8-s + (1.16 + 1.38i)10-s + (−0.487 − 0.487i)11-s + (−0.986 + 0.161i)13-s − 1.19i·14-s + 1.95·16-s + (−0.571 + 0.571i)17-s + (−1.13 − 1.13i)19-s + (1.47 + 1.75i)20-s + (−0.884 − 0.884i)22-s + (1.06 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.26158 + 0.291993i\)
\(L(\frac12)\) \(\approx\) \(4.26158 + 0.291993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 1.71i)T \)
13 \( 1 + (3.55 - 0.582i)T \)
good2 \( 1 - 2.56T + 2T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.61i)T + 11iT^{2} \)
17 \( 1 + (2.35 - 2.35i)T - 17iT^{2} \)
19 \( 1 + (4.94 + 4.94i)T + 19iT^{2} \)
23 \( 1 + (-5.08 - 5.08i)T + 23iT^{2} \)
29 \( 1 - 2.47iT - 29T^{2} \)
31 \( 1 + (-2.12 + 2.12i)T - 31iT^{2} \)
37 \( 1 + 5.15iT - 37T^{2} \)
41 \( 1 + (-1.37 + 1.37i)T - 41iT^{2} \)
43 \( 1 + (-3.62 - 3.62i)T + 43iT^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + (4.06 - 4.06i)T - 53iT^{2} \)
59 \( 1 + (3.97 - 3.97i)T - 59iT^{2} \)
61 \( 1 + 7.32T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (10.3 - 10.3i)T - 71iT^{2} \)
73 \( 1 + 6.48T + 73T^{2} \)
79 \( 1 - 3.78iT - 79T^{2} \)
83 \( 1 + 8.81iT - 83T^{2} \)
89 \( 1 + (-3.75 + 3.75i)T - 89iT^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−10.89844927533890931913342919230, −10.38517213169966488994023688566, −9.058475609952371013779239524130, −7.41569158602583707565956611703, −6.89990397459546303094375333323, −5.99858520426250031247605727775, −5.09912198925896088759864450142, −4.15245788590288413824704807420, −3.02701418944418739053875166745, −2.16544036290741713884026080155, 2.06134324006775529035609733558, 2.82503844922890995187007420223, 4.51703997861297962653307855168, 4.87119473059438400234487385417, 5.88116677876088519177778563470, 6.59531698297202524569124069880, 7.78562011224552459695812033694, 8.936997930070361125338264901157, 10.03486841862758558579149586600, 10.92009232171053664875391889098

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