Properties

Label 2-585-65.38-c0-0-0
Degree $2$
Conductor $585$
Sign $0.584 + 0.811i$
Analytic cond. $0.291953$
Root an. cond. $0.540326$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 1.30i)2-s + 2.41i·4-s + (0.923 − 0.382i)5-s + (1.84 − 1.84i)8-s + (−1.70 − 0.707i)10-s + 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (0.923 + 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s − 1.84i·26-s + (1.30 + 1.30i)32-s + (1.00 − 2.41i)40-s − 0.765i·41-s + ⋯
L(s)  = 1  + (−1.30 − 1.30i)2-s + 2.41i·4-s + (0.923 − 0.382i)5-s + (1.84 − 1.84i)8-s + (−1.70 − 0.707i)10-s + 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (0.923 + 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s − 1.84i·26-s + (1.30 + 1.30i)32-s + (1.00 − 2.41i)40-s − 0.765i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(0.291953\)
Root analytic conductor: \(0.540326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :0),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5680328029\)
\(L(\frac12)\) \(\approx\) \(0.5680328029\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 1.84iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 0.765iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 0.765T + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.765iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.39916256938488087519610781854, −10.02986158880638050593872769127, −9.189510731008228967679438582306, −8.655214474780048353855139455041, −7.49277186874296420089037597519, −6.59483893430629035520582645665, −4.95187547651253930837016627383, −3.80276706457507418806013034548, −2.27312621645193045876677843939, −1.59874041401367885536930546194, 1.23527477376490119975142196625, 3.10246286632800983312555546459, 5.18515876633768035895774300738, 6.10427129475741550000436138089, 6.35638345925590981295990084040, 7.67026016905902067042344895413, 8.411154622757731180766813334356, 9.087160265251616816100968054184, 9.946169014876179928628885276373, 10.73860052340098912597415889779

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