Properties

Label 2-252-12.11-c3-0-29
Degree $2$
Conductor $252$
Sign $0.909 + 0.415i$
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  + (2.45 + 1.41i)2-s + (4.02 + 6.91i)4-s − 16.9i·5-s − 7i·7-s + (0.100 + 22.6i)8-s + (23.8 − 41.4i)10-s − 13.3·11-s + 76.5·13-s + (9.87 − 17.1i)14-s + (−31.6 + 55.6i)16-s − 99.7i·17-s − 121. i·19-s + (117. − 68.0i)20-s + (−32.7 − 18.8i)22-s + 151.·23-s + ⋯
L(s)  = 1  + (0.866 + 0.498i)2-s + (0.502 + 0.864i)4-s − 1.51i·5-s − 0.377i·7-s + (0.00445 + 0.999i)8-s + (0.754 − 1.31i)10-s − 0.366·11-s + 1.63·13-s + (0.188 − 0.327i)14-s + (−0.494 + 0.868i)16-s − 1.42i·17-s − 1.46i·19-s + (1.30 − 0.760i)20-s + (−0.317 − 0.182i)22-s + 1.37·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.909 + 0.415i$
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: \((2,\ 252,\ (\ :3/2),\ 0.909 + 0.415i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.037944280\)
\(L(\frac12)\) \(\approx\) \(3.037944280\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.45 - 1.41i)T \)
3 \( 1 \)
7 \( 1 + 7iT \)
good5 \( 1 + 16.9iT - 125T^{2} \)
11 \( 1 + 13.3T + 1.33e3T^{2} \)
13 \( 1 - 76.5T + 2.19e3T^{2} \)
17 \( 1 + 99.7iT - 4.91e3T^{2} \)
19 \( 1 + 121. iT - 6.85e3T^{2} \)
23 \( 1 - 151.T + 1.21e4T^{2} \)
29 \( 1 + 197. iT - 2.43e4T^{2} \)
31 \( 1 - 268. iT - 2.97e4T^{2} \)
37 \( 1 - 19.2T + 5.06e4T^{2} \)
41 \( 1 - 129. iT - 6.89e4T^{2} \)
43 \( 1 - 490. iT - 7.95e4T^{2} \)
47 \( 1 - 160.T + 1.03e5T^{2} \)
53 \( 1 + 126. iT - 1.48e5T^{2} \)
59 \( 1 + 285.T + 2.05e5T^{2} \)
61 \( 1 + 467.T + 2.26e5T^{2} \)
67 \( 1 + 323. iT - 3.00e5T^{2} \)
71 \( 1 + 673.T + 3.57e5T^{2} \)
73 \( 1 + 338.T + 3.89e5T^{2} \)
79 \( 1 - 869. iT - 4.93e5T^{2} \)
83 \( 1 - 532.T + 5.71e5T^{2} \)
89 \( 1 - 140. iT - 7.04e5T^{2} \)
97 \( 1 - 652.T + 9.12e5T^{2} \)
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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

−11.67651637176744157607155642632, −10.97292327575493332668357181325, −9.226901248647620965959380906412, −8.576789612259652707504250080242, −7.49881769383431576303216516521, −6.34592178769509275327731524817, −5.08261949086004010696289637493, −4.53616644650326031747530448482, −3.04495402556281078567470048472, −0.997995566748957443587308279678, 1.73891157986315579112247832405, 3.13527339043049981109433382845, 3.89785790606092090683870903112, 5.73444351906465791004204968939, 6.28130125830159478769417251989, 7.44527529629927210037559275461, 8.852953368704579867478496748667, 10.46170020472003302699212965280, 10.62299023104127186815344210104, 11.56526275818581494285091291255

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