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Properties

Label 2-252-12.11-c3-0-32
Degree $2$
Conductor $252$
Sign $-0.104 - 0.994i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
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  + (−1.42 − 2.44i)2-s + (−3.91 + 6.97i)4-s − 15.7i·5-s + 7i·7-s + (22.6 − 0.420i)8-s + (−38.5 + 22.5i)10-s − 23.5·11-s − 37.7·13-s + (17.0 − 10.0i)14-s + (−33.3 − 54.6i)16-s − 31.9i·17-s − 28.8i·19-s + (110. + 61.7i)20-s + (33.6 + 57.3i)22-s − 50.8·23-s + ⋯
L(s)  = 1  + (−0.505 − 0.862i)2-s + (−0.489 + 0.872i)4-s − 1.41i·5-s + 0.377i·7-s + (0.999 − 0.0186i)8-s + (−1.21 + 0.713i)10-s − 0.644·11-s − 0.804·13-s + (0.326 − 0.191i)14-s + (−0.521 − 0.853i)16-s − 0.455i·17-s − 0.348i·19-s + (1.23 + 0.690i)20-s + (0.325 + 0.556i)22-s − 0.461·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.104 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.002121766207\)
\(L(\frac12)\) \(\approx\) \(0.002121766207\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.42 + 2.44i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 + 15.7iT - 125T^{2} \)
11 \( 1 + 23.5T + 1.33e3T^{2} \)
13 \( 1 + 37.7T + 2.19e3T^{2} \)
17 \( 1 + 31.9iT - 4.91e3T^{2} \)
19 \( 1 + 28.8iT - 6.85e3T^{2} \)
23 \( 1 + 50.8T + 1.21e4T^{2} \)
29 \( 1 - 205. iT - 2.43e4T^{2} \)
31 \( 1 - 176. iT - 2.97e4T^{2} \)
37 \( 1 + 338.T + 5.06e4T^{2} \)
41 \( 1 - 77.7iT - 6.89e4T^{2} \)
43 \( 1 - 463. iT - 7.95e4T^{2} \)
47 \( 1 - 352.T + 1.03e5T^{2} \)
53 \( 1 + 155. iT - 1.48e5T^{2} \)
59 \( 1 + 761.T + 2.05e5T^{2} \)
61 \( 1 + 40.2T + 2.26e5T^{2} \)
67 \( 1 - 838. iT - 3.00e5T^{2} \)
71 \( 1 - 70.1T + 3.57e5T^{2} \)
73 \( 1 - 726.T + 3.89e5T^{2} \)
79 \( 1 + 992. iT - 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + 1.59e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−10.88060922332418626953068472330, −9.844016713504559285521180317395, −9.004520701751747024953903241704, −8.338903589058343201848687738759, −7.24090098734340911709070230393, −5.28396980020401986869220256161, −4.58295826513405785796893964283, −2.93891869927736736667030542694, −1.48871983889555548568447284296, −0.000990753402413263835330366652, 2.27877556937940540856733991497, 3.96472379924682693743924137734, 5.48083320312262386794162524076, 6.51796296865278970222810913356, 7.38054299893134547579134717359, 8.072780433436922760193648881796, 9.522653930428145406091879142220, 10.36516204513867669663794716017, 10.86960151726477043561364523978, 12.24950291147342165565635984676

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