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Properties

Label 2-12e3-9.5-c2-0-1
Degree $2$
Conductor $1728$
Sign $-0.797 - 0.603i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0440 − 0.0254i)5-s + (−4.52 − 7.84i)7-s + (−3.29 + 1.90i)11-s + (−0.216 + 0.375i)13-s + 26.2i·17-s + 34.2·19-s + (−29.9 − 17.3i)23-s + (−12.4 − 21.6i)25-s + (14.0 − 8.10i)29-s + (17.1 − 29.7i)31-s + 0.460i·35-s − 29.2·37-s + (−48.7 − 28.1i)41-s + (−3.94 − 6.83i)43-s + (−33.4 + 19.3i)47-s + ⋯
L(s)  = 1  + (−0.00880 − 0.00508i)5-s + (−0.647 − 1.12i)7-s + (−0.299 + 0.172i)11-s + (−0.0166 + 0.0288i)13-s + 1.54i·17-s + 1.80·19-s + (−1.30 − 0.752i)23-s + (−0.499 − 0.865i)25-s + (0.483 − 0.279i)29-s + (0.553 − 0.959i)31-s + 0.0131i·35-s − 0.791·37-s + (−1.18 − 0.685i)41-s + (−0.0917 − 0.158i)43-s + (−0.711 + 0.410i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.797 - 0.603i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.797 - 0.603i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1787182130\)
\(L(\frac12)\) \(\approx\) \(0.1787182130\)
\(L(2)\) 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 \( 1 + (0.0440 + 0.0254i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.52 + 7.84i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.29 - 1.90i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.216 - 0.375i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.2iT - 289T^{2} \)
19 \( 1 - 34.2T + 361T^{2} \)
23 \( 1 + (29.9 + 17.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-14.0 + 8.10i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-17.1 + 29.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 29.2T + 1.36e3T^{2} \)
41 \( 1 + (48.7 + 28.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (3.94 + 6.83i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (33.4 - 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 50.5iT - 2.80e3T^{2} \)
59 \( 1 + (-8.54 - 4.93i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.5 - 63.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (12.6 - 21.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 97.8iT - 5.04e3T^{2} \)
73 \( 1 + 77.0T + 5.32e3T^{2} \)
79 \( 1 + (-42.1 - 72.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (40.6 - 23.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 108. iT - 7.92e3T^{2} \)
97 \( 1 + (32.4 + 56.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.794345583391832805687310580829, −8.481408994617824133103130154430, −7.905422808973863492526314013044, −7.05691157059954381143958225641, −6.30601812720598028907360552346, −5.48546445126370930243953741826, −4.23438730490430086604616209167, −3.74257723936243326658040572850, −2.55149372343305084463616915866, −1.21398193733308094846140476665, 0.04882074530511757047563925387, 1.64055709549905143855350142156, 2.94535125272394237060220547327, 3.39323568092044107735177015431, 5.04227865262795539461124514175, 5.39503078238555270792926679517, 6.39819118330112904439273692086, 7.24106196459181097111774908282, 8.041846867701419025481434425372, 8.933540781830903235904675387992

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