Properties

Label 2-864-288.13-c1-0-25
Degree $2$
Conductor $864$
Sign $0.279 + 0.960i$
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  + (0.238 − 1.39i)2-s + (−1.88 − 0.663i)4-s + (−0.0805 + 0.611i)5-s + (2.18 + 0.584i)7-s + (−1.37 + 2.47i)8-s + (0.833 + 0.257i)10-s + (0.118 − 0.154i)11-s + (0.882 − 0.677i)13-s + (1.33 − 2.90i)14-s + (3.11 + 2.50i)16-s − 2.51i·17-s + (1.37 + 0.568i)19-s + (0.557 − 1.10i)20-s + (−0.186 − 0.201i)22-s + (2.32 − 0.624i)23-s + ⋯
L(s)  = 1  + (0.168 − 0.985i)2-s + (−0.943 − 0.331i)4-s + (−0.0360 + 0.273i)5-s + (0.824 + 0.220i)7-s + (−0.485 + 0.874i)8-s + (0.263 + 0.0815i)10-s + (0.0356 − 0.0465i)11-s + (0.244 − 0.187i)13-s + (0.356 − 0.775i)14-s + (0.779 + 0.626i)16-s − 0.609i·17-s + (0.315 + 0.130i)19-s + (0.124 − 0.246i)20-s + (−0.0398 − 0.0430i)22-s + (0.485 − 0.130i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35972 - 1.02076i\)
\(L(\frac12)\) \(\approx\) \(1.35972 - 1.02076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.238 + 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (0.0805 - 0.611i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-2.18 - 0.584i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.118 + 0.154i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (-0.882 + 0.677i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 + 2.51iT - 17T^{2} \)
19 \( 1 + (-1.37 - 0.568i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.32 + 0.624i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-9.87 + 1.29i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (-1.77 + 3.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.25 - 2.59i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.164 + 0.0442i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.05 + 3.98i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-6.51 + 3.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.44 + 5.90i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.957 - 7.27i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (11.8 - 1.56i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (-2.90 - 3.78i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (2.34 - 2.34i)T - 71iT^{2} \)
73 \( 1 + (8.46 + 8.46i)T + 73iT^{2} \)
79 \( 1 + (-8.87 + 5.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.750 - 5.70i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-9.23 + 9.23i)T - 89iT^{2} \)
97 \( 1 + (-8.80 - 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.33598065979673410422147553899, −9.170205358516131603390680500429, −8.558920816619535767497428009228, −7.62244119748561314030799673338, −6.40638895794039983412593601198, −5.23965899584021918826236372551, −4.60758920324134761862024474983, −3.36015656687471341617055439233, −2.41628864005323443350791859929, −1.05026878707371286488534296766, 1.19106397292009139329758816180, 3.13559951377210802658252474881, 4.43985677426228884281248573658, 4.94259454689620682767815475806, 6.06005039100887154101453345828, 6.90487040105133359249720527692, 7.79467553314860879890878043739, 8.535389390382889912370406763552, 9.141644568718249466938689650582, 10.28690611902286381367830120691

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