Properties

Label 2-864-36.11-c1-0-10
Degree $2$
Conductor $864$
Sign $0.213 + 0.976i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (3.01 − 1.73i)5-s + (−3.12 − 1.80i)7-s + (1.32 − 2.29i)11-s + (2.36 + 4.09i)13-s − 1.79i·17-s − 4.55i·19-s + (−0.377 − 0.653i)23-s + (3.54 − 6.13i)25-s + (−7.19 − 4.15i)29-s + (1.94 − 1.12i)31-s − 12.5·35-s − 3.98·37-s + (5.57 − 3.22i)41-s + (7.60 + 4.39i)43-s + (−1.37 + 2.38i)47-s + ⋯
L(s)  = 1  + (1.34 − 0.777i)5-s + (−1.18 − 0.681i)7-s + (0.399 − 0.691i)11-s + (0.655 + 1.13i)13-s − 0.434i·17-s − 1.04i·19-s + (−0.0787 − 0.136i)23-s + (0.708 − 1.22i)25-s + (−1.33 − 0.771i)29-s + (0.348 − 0.201i)31-s − 2.11·35-s − 0.655·37-s + (0.871 − 0.502i)41-s + (1.15 + 0.669i)43-s + (−0.201 + 0.348i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31938 - 1.06165i\)
\(L(\frac12)\) \(\approx\) \(1.31938 - 1.06165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-3.01 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.12 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.36 - 4.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.79iT - 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 + (0.377 + 0.653i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.19 + 4.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.94 + 1.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.98T + 37T^{2} \)
41 \( 1 + (-5.57 + 3.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.60 - 4.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.37 - 2.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 + (1.36 + 2.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.19 + 2.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.78 - 5.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.0730T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + (-4.51 - 2.60i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.244 - 0.424i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.41iT - 89T^{2} \)
97 \( 1 + (-7.21 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.556948462959980266253828302596, −9.441616861616118070630254222901, −8.621115480275835560346281916006, −7.21025865046260823544178117992, −6.35914263052040071912611390095, −5.82437296438919580020813457264, −4.60100730362483384766743739851, −3.58377013042574429107476053243, −2.22263297022784838647546993526, −0.838897518136814746470972743790, 1.74332154917715879185751655821, 2.84383201023571177248486209233, 3.71839931525773219753202058260, 5.48978903508518166263088928430, 5.98475954063916217368475334696, 6.65770172676126403050348374665, 7.70972246326939462567357318852, 8.981363026810218401361891989001, 9.549435684579085421803245464983, 10.31088696500775294130925629290

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