Properties

Label 2-864-32.13-c1-0-26
Degree $2$
Conductor $864$
Sign $-0.387 + 0.921i$
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  + (−0.274 − 1.38i)2-s + (−1.84 + 0.761i)4-s + (−3.91 + 1.61i)5-s + (−0.876 + 0.876i)7-s + (1.56 + 2.35i)8-s + (3.31 + 4.98i)10-s + (0.423 + 1.02i)11-s + (0.652 + 0.270i)13-s + (1.45 + 0.975i)14-s + (2.84 − 2.81i)16-s − 4.86i·17-s + (2.72 + 1.12i)19-s + (5.99 − 5.97i)20-s + (1.30 − 0.867i)22-s + (−5.27 − 5.27i)23-s + ⋯
L(s)  = 1  + (−0.194 − 0.980i)2-s + (−0.924 + 0.380i)4-s + (−1.74 + 0.724i)5-s + (−0.331 + 0.331i)7-s + (0.552 + 0.833i)8-s + (1.04 + 1.57i)10-s + (0.127 + 0.307i)11-s + (0.181 + 0.0749i)13-s + (0.389 + 0.260i)14-s + (0.710 − 0.704i)16-s − 1.18i·17-s + (0.625 + 0.258i)19-s + (1.34 − 1.33i)20-s + (0.277 − 0.184i)22-s + (−1.09 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.387 + 0.921i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315459 - 0.474883i\)
\(L(\frac12)\) \(\approx\) \(0.315459 - 0.474883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.274 + 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (3.91 - 1.61i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.876 - 0.876i)T - 7iT^{2} \)
11 \( 1 + (-0.423 - 1.02i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.652 - 0.270i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.86iT - 17T^{2} \)
19 \( 1 + (-2.72 - 1.12i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.27 + 5.27i)T + 23iT^{2} \)
29 \( 1 + (2.12 - 5.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 + (-7.76 + 3.21i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.84 + 3.84i)T + 41iT^{2} \)
43 \( 1 + (1.66 + 4.00i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.62iT - 47T^{2} \)
53 \( 1 + (-1.11 - 2.69i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-12.6 + 5.25i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.82 + 6.82i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.48 + 8.41i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-6.28 + 6.28i)T - 71iT^{2} \)
73 \( 1 + (0.766 + 0.766i)T + 73iT^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + (-14.5 - 6.02i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-3.65 + 3.65i)T - 89iT^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−10.03470245762733158134701334068, −9.158306859896287657648044347127, −8.235052669659046046346353389522, −7.56052748783780884455198275767, −6.70163608346886326457898129047, −5.14190220250163640026476829382, −4.08868968136458002662038408623, −3.41551100515444770703705753701, −2.42957618222193601589960311900, −0.40093695928635205781446954611, 0.981030437599880388601862106825, 3.66951070009732913298162475770, 4.05377665172522415771316862147, 5.17623077283601885524658889441, 6.18461903948078181399326478392, 7.23332385552793283137438506788, 7.978270981799610511261083718320, 8.361274914429933531657971805948, 9.360050111389379045363797128645, 10.21838604638131212377122440647

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