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Properties

Label 2-12e3-9.5-c2-0-35
Degree $2$
Conductor $1728$
Sign $0.0415 + 0.999i$
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  + (3.44 + 1.98i)5-s + (1.80 + 3.12i)7-s + (−11.8 + 6.83i)11-s + (8.96 − 15.5i)13-s − 21.6i·17-s − 23.4·19-s + (13.0 + 7.54i)23-s + (−4.59 − 7.96i)25-s + (−20.4 + 11.8i)29-s + (23.5 − 40.7i)31-s + 14.3i·35-s − 54.6·37-s + (−24.6 − 14.2i)41-s + (−23.8 − 41.3i)43-s + (30.5 − 17.6i)47-s + ⋯
L(s)  = 1  + (0.688 + 0.397i)5-s + (0.257 + 0.446i)7-s + (−1.07 + 0.621i)11-s + (0.689 − 1.19i)13-s − 1.27i·17-s − 1.23·19-s + (0.568 + 0.327i)23-s + (−0.183 − 0.318i)25-s + (−0.706 + 0.407i)29-s + (0.758 − 1.31i)31-s + 0.409i·35-s − 1.47·37-s + (−0.601 − 0.347i)41-s + (−0.555 − 0.961i)43-s + (0.649 − 0.375i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0415 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0415 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.0415 + 0.999i$
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: \((2,\ 1728,\ (\ :1),\ 0.0415 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.441474731\)
\(L(\frac12)\) \(\approx\) \(1.441474731\)
\(L(2)\) 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 \( 1 + (-3.44 - 1.98i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.80 - 3.12i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (11.8 - 6.83i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.96 + 15.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 21.6iT - 289T^{2} \)
19 \( 1 + 23.4T + 361T^{2} \)
23 \( 1 + (-13.0 - 7.54i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (20.4 - 11.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23.5 + 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 54.6T + 1.36e3T^{2} \)
41 \( 1 + (24.6 + 14.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (23.8 + 41.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-30.5 + 17.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 65.0iT - 2.80e3T^{2} \)
59 \( 1 + (-76.0 - 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.46 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-1.55 + 2.69i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 49.5iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 + (-18.7 - 32.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (14.6 - 8.48i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 14.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.1 + 41.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.848404716030149122950672997856, −8.240695672160734326765394006198, −7.32092979864148885779442044357, −6.56917664208888328040396842183, −5.42713072178939277976505891504, −5.23116059622672380825228091930, −3.81527237550958166799156742340, −2.65749646502157407507744756790, −2.05304782125940523301585241491, −0.36762567240788976085118012160, 1.28353801180217620337091925667, 2.13604492219071260463169556821, 3.46656430786351724543778848938, 4.40504385151518025644764662712, 5.26257932081889872850392645910, 6.15866040750163160784960608276, 6.76373972603742844884428675447, 7.943444039395177226456900386066, 8.584092524838038379807659848615, 9.158629536981413392401980290611

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