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Properties

Label 2-12e3-36.7-c2-0-32
Degree $2$
Conductor $1728$
Sign $-0.208 + 0.978i$
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  + (−3.01 − 5.22i)5-s + (10.2 + 5.90i)7-s + (5.28 + 3.05i)11-s + (−7.44 − 12.9i)13-s − 26.6·17-s + 9.45i·19-s + (17.2 − 9.96i)23-s + (−5.70 + 9.88i)25-s + (22.3 − 38.6i)29-s + (5.42 − 3.13i)31-s − 71.3i·35-s + 6.65·37-s + (−8.82 − 15.2i)41-s + (−20.2 − 11.7i)43-s + (36.4 + 21.0i)47-s + ⋯
L(s)  = 1  + (−0.603 − 1.04i)5-s + (1.46 + 0.844i)7-s + (0.480 + 0.277i)11-s + (−0.572 − 0.992i)13-s − 1.57·17-s + 0.497i·19-s + (0.750 − 0.433i)23-s + (−0.228 + 0.395i)25-s + (0.769 − 1.33i)29-s + (0.174 − 0.101i)31-s − 2.03i·35-s + 0.179·37-s + (−0.215 − 0.372i)41-s + (−0.471 − 0.272i)43-s + (0.775 + 0.447i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.208 + 0.978i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.208 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.601988696\)
\(L(\frac12)\) \(\approx\) \(1.601988696\)
\(L(2)\) 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 \( 1 + (3.01 + 5.22i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-10.2 - 5.90i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.28 - 3.05i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.44 + 12.9i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 26.6T + 289T^{2} \)
19 \( 1 - 9.45iT - 361T^{2} \)
23 \( 1 + (-17.2 + 9.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.3 + 38.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-5.42 + 3.13i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 6.65T + 1.36e3T^{2} \)
41 \( 1 + (8.82 + 15.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (20.2 + 11.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.4 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 51.6T + 2.80e3T^{2} \)
59 \( 1 + (32.9 - 18.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.3 + 78.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (53.4 - 30.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 39.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.0T + 5.32e3T^{2} \)
79 \( 1 + (77.9 + 45.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-102. - 59.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 14.4T + 7.92e3T^{2} \)
97 \( 1 + (-67.5 + 117. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.716779642926256541798451865952, −8.238216852629852489760896456500, −7.63445334275366650910145828942, −6.46424685328063860439356048077, −5.41812198457341122965138631519, −4.70427091962942745534016216705, −4.26341423632477661587967331752, −2.67448695060971532711790239060, −1.69224713044837702362012688491, −0.45046221253912567995158382555, 1.22647464713998315601285775920, 2.36786377089504614721729966092, 3.52199931009206927810265480907, 4.46649098260700718468392233070, 4.94632072701789661310179622474, 6.54253546475006646828468381568, 6.99928979909334634852401091132, 7.59381462557953076115061533636, 8.575084263501586407430368434319, 9.193914393442675020514520209860

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