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Properties

Label 2-252-21.17-c3-0-1
Degree $2$
Conductor $252$
Sign $0.113 - 0.993i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.113 - 0.993i$
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: \((2,\ 252,\ (\ :3/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.847746982\)
\(L(\frac12)\) \(\approx\) \(1.847746982\)
\(L(\frac{5}{2})\) 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

−11.75798773043874763119576499788, −10.85329495219592311356258533550, −10.04260041315534440539190295688, −8.844417276458889650683189152404, −7.987671943190253735108158922806, −6.74196771691597992281688509085, −5.77126948925161830619266160362, −4.59181925554004710838435332366, −2.98978643198715327900211427895, −1.71473084293138441761314249989, 0.75326318466503477988626633132, 2.26707478775007110286628792219, 4.14512184157193973702997030991, 5.04997597655910313756933795966, 6.23747874378651042158400753870, 7.60761839443753877412503763268, 8.350340226792314999269982908911, 9.525038574547649869160996787631, 10.36372299065071568747794592364, 11.42408841294013077486007790551

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