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Properties

Label 2-864-96.11-c1-0-18
Degree $2$
Conductor $864$
Sign $0.943 - 0.331i$
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  + (−0.948 − 1.04i)2-s + (−0.200 + 1.98i)4-s + (1.78 + 0.738i)5-s + (−0.622 + 0.622i)7-s + (2.27 − 1.67i)8-s + (−0.916 − 2.57i)10-s + (0.156 + 0.0648i)11-s + (1.19 + 2.89i)13-s + (1.24 + 0.0625i)14-s + (−3.91 − 0.799i)16-s + 4.81·17-s + (−3.67 + 1.52i)19-s + (−1.82 + 3.39i)20-s + (−0.0804 − 0.225i)22-s + (1.56 − 1.56i)23-s + ⋯
L(s)  = 1  + (−0.670 − 0.741i)2-s + (−0.100 + 0.994i)4-s + (0.797 + 0.330i)5-s + (−0.235 + 0.235i)7-s + (0.805 − 0.592i)8-s + (−0.289 − 0.812i)10-s + (0.0471 + 0.0195i)11-s + (0.332 + 0.802i)13-s + (0.332 + 0.0167i)14-s + (−0.979 − 0.199i)16-s + 1.16·17-s + (−0.843 + 0.349i)19-s + (−0.408 + 0.760i)20-s + (−0.0171 − 0.0481i)22-s + (0.326 − 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18171 + 0.201279i\)
\(L(\frac12)\) \(\approx\) \(1.18171 + 0.201279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.948 + 1.04i)T \)
3 \( 1 \)
good5 \( 1 + (-1.78 - 0.738i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.622 - 0.622i)T - 7iT^{2} \)
11 \( 1 + (-0.156 - 0.0648i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.19 - 2.89i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.81T + 17T^{2} \)
19 \( 1 + (3.67 - 1.52i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.56 + 1.56i)T - 23iT^{2} \)
29 \( 1 + (-1.08 - 2.61i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 2.74iT - 31T^{2} \)
37 \( 1 + (1.96 - 4.74i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.95 - 3.95i)T + 41iT^{2} \)
43 \( 1 + (0.627 - 1.51i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.33iT - 47T^{2} \)
53 \( 1 + (-0.182 + 0.441i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.02 + 7.30i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-11.6 + 4.83i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.45 - 10.7i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.22 + 6.22i)T + 71iT^{2} \)
73 \( 1 + (0.573 - 0.573i)T - 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (4.21 + 10.1i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.68 + 6.68i)T - 89iT^{2} \)
97 \( 1 + 5.92T + 97T^{2} \)
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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

−10.12982076430597719724193546334, −9.539555193152296125023406864495, −8.727483675895224813076815431808, −7.899850075226919795383890917210, −6.78372421717162264123097982807, −6.05620735666079281265220122372, −4.69526304954578802815714021777, −3.51307484101402942534155184246, −2.49405254002100261149620729325, −1.39328415517967529982101774297, 0.802882386249448544057702941009, 2.18779935404564567967099909541, 3.83317188065147153994868901650, 5.28173802796474697148356275050, 5.73270591752173066823943568692, 6.72016510862572863827970951660, 7.61360727570156934877948387920, 8.460147365162680371541864297126, 9.236048321637821585333842723964, 10.02857509274586468384739093991

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