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Properties

Label 2-42e2-7.2-c3-0-5
Degree $2$
Conductor $1764$
Sign $-0.386 - 0.922i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.22 + 10.7i)5-s + (−25.5 − 44.2i)11-s − 37.2·13-s + (−11.1 − 19.2i)17-s + (27.1 − 47.0i)19-s + (88.4 − 153. i)23-s + (−14.9 − 25.9i)25-s − 61.0·29-s + (159. + 277. i)31-s + (157. − 272. i)37-s − 206.·41-s + 339.·43-s + (−71.0 + 123. i)47-s + (155. + 268. i)53-s + 636.·55-s + ⋯
L(s)  = 1  + (−0.556 + 0.964i)5-s + (−0.700 − 1.21i)11-s − 0.794·13-s + (−0.158 − 0.274i)17-s + (0.328 − 0.568i)19-s + (0.801 − 1.38i)23-s + (−0.119 − 0.207i)25-s − 0.391·29-s + (0.926 + 1.60i)31-s + (0.699 − 1.21i)37-s − 0.784·41-s + 1.20·43-s + (−0.220 + 0.381i)47-s + (0.401 + 0.695i)53-s + 1.55·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8312847779\)
\(L(\frac12)\) \(\approx\) \(0.8312847779\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (6.22 - 10.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (25.5 + 44.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 37.2T + 2.19e3T^{2} \)
17 \( 1 + (11.1 + 19.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-27.1 + 47.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-88.4 + 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 61.0T + 2.43e4T^{2} \)
31 \( 1 + (-159. - 277. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-157. + 272. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 206.T + 6.89e4T^{2} \)
43 \( 1 - 339.T + 7.95e4T^{2} \)
47 \( 1 + (71.0 - 123. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-155. - 268. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-140. - 243. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (271. - 470. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-239. - 415. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + (-119. - 207. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (580. - 1.00e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 2.93T + 5.71e5T^{2} \)
89 \( 1 + (-639. + 1.10e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 79.0T + 9.12e5T^{2} \)
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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

−9.020825292116068405791860680564, −8.437538582235923821646261595341, −7.39708565209264171116465815377, −7.02400118259483380959646769671, −6.02117657769827667713715245093, −5.11645454525973998436491332834, −4.18567407459799671968933352342, −2.88309018104490987649317037707, −2.75864214222609468147145653552, −0.847854176775164903666087219162, 0.22630158556923908859196100968, 1.47151720404592913041475076776, 2.54670224662013117933191134006, 3.78257080788089490356439300538, 4.71770880556771709888292946716, 5.14250588282257076259255072644, 6.25615015023341195355779172442, 7.46419716451246489371392965461, 7.73819882065972228142847591767, 8.630611907983603576364405377335

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