↑

Properties

Label 2-252-63.41-c3-0-8
Degree $2$
Conductor $252$
Sign $0.988 - 0.151i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.316 − 5.18i)3-s + (−5.49 + 9.52i)5-s + (−18.3 − 2.66i)7-s + (−26.8 + 3.28i)9-s + (13.8 − 7.98i)11-s + (77.4 + 44.7i)13-s + (51.1 + 25.4i)15-s + 106.·17-s + 8.31i·19-s + (−8.02 + 95.8i)21-s + (−123. − 71.5i)23-s + (2.06 + 3.58i)25-s + (25.4 + 137. i)27-s + (129. − 74.8i)29-s + (37.1 + 21.4i)31-s + ⋯
L(s)  = 1  + (−0.0608 − 0.998i)3-s + (−0.491 + 0.851i)5-s + (−0.989 − 0.143i)7-s + (−0.992 + 0.121i)9-s + (0.378 − 0.218i)11-s + (1.65 + 0.953i)13-s + (0.879 + 0.438i)15-s + 1.52·17-s + 0.100i·19-s + (−0.0833 + 0.996i)21-s + (−1.12 − 0.648i)23-s + (0.0165 + 0.0286i)25-s + (0.181 + 0.983i)27-s + (0.830 − 0.479i)29-s + (0.215 + 0.124i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.413287678\)
\(L(\frac12)\) \(\approx\) \(1.413287678\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.316 + 5.18i)T \)
7 \( 1 + (18.3 + 2.66i)T \)
good5 \( 1 + (5.49 - 9.52i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-13.8 + 7.98i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-77.4 - 44.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 106.T + 4.91e3T^{2} \)
19 \( 1 - 8.31iT - 6.85e3T^{2} \)
23 \( 1 + (123. + 71.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-129. + 74.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-37.1 - 21.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 390.T + 5.06e4T^{2} \)
41 \( 1 + (172. - 298. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-28.5 - 49.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (8.13 + 14.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 445. iT - 1.48e5T^{2} \)
59 \( 1 + (193. - 334. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-420. + 242. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (251. - 436. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 751. iT - 3.57e5T^{2} \)
73 \( 1 + 507. iT - 3.89e5T^{2} \)
79 \( 1 + (-381. - 660. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-607. - 1.05e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 425.T + 7.04e5T^{2} \)
97 \( 1 + (494. - 285. i)T + (4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.70225477784970095706679891308, −10.88488667043979867047254729664, −9.738951413075651961186081282016, −8.488251768248463754763820595916, −7.58797087421629460006544758032, −6.42582182360055387319088420048, −6.12840105809382872354939850008, −3.87993161924237510037015975827, −2.89985176413613375582652289489, −1.08770208041322501447214441930, 0.73538142866861580704372145854, 3.26524695617570441632051893317, 4.00935843766954828126028825664, 5.40255033905716142432314423228, 6.21335993645965003843970838680, 7.985199061584868400407776318129, 8.723275903353815280379177596744, 9.729541703360589288444300806007, 10.42598626462510466622944431919, 11.63627485258608530680203231832

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