Properties

Label 2-252-21.17-c3-0-4
Degree $2$
Conductor $252$
Sign $0.898 + 0.438i$
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  + (−4.36 − 7.56i)5-s + (14.4 + 11.5i)7-s + (7.60 + 4.39i)11-s + 11.8i·13-s + (22.2 − 38.6i)17-s + (10.0 − 5.82i)19-s + (123. − 71.3i)23-s + (24.3 − 42.1i)25-s − 234. i·29-s + (252. + 145. i)31-s + (23.9 − 160. i)35-s + (44.4 + 76.9i)37-s + 145.·41-s + 144.·43-s + (−120. − 208. i)47-s + ⋯
L(s)  = 1  + (−0.390 − 0.676i)5-s + (0.782 + 0.622i)7-s + (0.208 + 0.120i)11-s + 0.252i·13-s + (0.318 − 0.550i)17-s + (0.121 − 0.0703i)19-s + (1.11 − 0.646i)23-s + (0.194 − 0.337i)25-s − 1.49i·29-s + (1.46 + 0.845i)31-s + (0.115 − 0.772i)35-s + (0.197 + 0.342i)37-s + 0.555·41-s + 0.512·43-s + (−0.372 − 0.646i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.898 + 0.438i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.867037217\)
\(L(\frac12)\) \(\approx\) \(1.867037217\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.4 - 11.5i)T \)
good5 \( 1 + (4.36 + 7.56i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.60 - 4.39i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 11.8iT - 2.19e3T^{2} \)
17 \( 1 + (-22.2 + 38.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-10.0 + 5.82i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-123. + 71.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 234. iT - 2.43e4T^{2} \)
31 \( 1 + (-252. - 145. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-44.4 - 76.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 145.T + 6.89e4T^{2} \)
43 \( 1 - 144.T + 7.95e4T^{2} \)
47 \( 1 + (120. + 208. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-263. - 152. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-3.54 + 6.13i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (149. - 86.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-243. + 421. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 653. iT - 3.57e5T^{2} \)
73 \( 1 + (99.0 + 57.1i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (147. + 255. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 877.T + 5.71e5T^{2} \)
89 \( 1 + (-710. - 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 738. iT - 9.12e5T^{2} \)
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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

−11.80021833727769543289217631636, −10.68198455182369293342268018342, −9.429364087759273932167984138656, −8.586434509977010470769449185899, −7.79292674278735029051100916855, −6.48059956849416865318556784470, −5.13830018265612590203636044525, −4.36870729287541225996474456809, −2.61533623480734461307672275936, −0.955814519366399086550285932077, 1.19641289905962759398958801183, 3.03571174662691110399937821537, 4.21252336694129147497075410536, 5.48701144324457212000685987475, 6.87573408523157358938316202506, 7.63335363689393266111771649316, 8.628674301001009542440299806272, 9.911260988702193425416323561403, 10.92478888420561270062172295100, 11.37405742193130628778538349263

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