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Properties

Label 2-252-63.41-c3-0-16
Degree $2$
Conductor $252$
Sign $-0.511 + 0.859i$
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  + (−5.19 − 0.205i)3-s + (2.34 − 4.05i)5-s + (−10.9 + 14.9i)7-s + (26.9 + 2.13i)9-s + (16.1 − 9.30i)11-s + (44.1 + 25.4i)13-s + (−12.9 + 20.5i)15-s − 112.·17-s − 111. i·19-s + (59.8 − 75.3i)21-s + (−124. − 71.8i)23-s + (51.5 + 89.2i)25-s + (−139. − 16.6i)27-s + (206. − 119. i)29-s + (−179. − 103. i)31-s + ⋯
L(s)  = 1  + (−0.999 − 0.0395i)3-s + (0.209 − 0.362i)5-s + (−0.590 + 0.807i)7-s + (0.996 + 0.0790i)9-s + (0.441 − 0.255i)11-s + (0.941 + 0.543i)13-s + (−0.223 + 0.354i)15-s − 1.60·17-s − 1.34i·19-s + (0.622 − 0.783i)21-s + (−1.12 − 0.651i)23-s + (0.412 + 0.713i)25-s + (−0.992 − 0.118i)27-s + (1.32 − 0.764i)29-s + (−1.04 − 0.602i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.511 + 0.859i$
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: \((2,\ 252,\ (\ :3/2),\ -0.511 + 0.859i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6026278213\)
\(L(\frac12)\) \(\approx\) \(0.6026278213\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.19 + 0.205i)T \)
7 \( 1 + (10.9 - 14.9i)T \)
good5 \( 1 + (-2.34 + 4.05i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-16.1 + 9.30i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-44.1 - 25.4i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 112.T + 4.91e3T^{2} \)
19 \( 1 + 111. iT - 6.85e3T^{2} \)
23 \( 1 + (124. + 71.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-206. + 119. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (179. + 103. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 227.T + 5.06e4T^{2} \)
41 \( 1 + (-133. + 230. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (170. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (111. + 193. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 547. iT - 1.48e5T^{2} \)
59 \( 1 + (43.9 - 76.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (312. - 180. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (372. - 645. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 135. iT - 3.57e5T^{2} \)
73 \( 1 + 467. iT - 3.89e5T^{2} \)
79 \( 1 + (-192. - 332. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-597. - 1.03e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 + (-1.07e3 + 617. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.40326977976263839639083845058, −10.49441001793400892071172509391, −9.229281913000938343746307202190, −8.667915788079717626224496215675, −6.82649365921194545324434120712, −6.28582927334780790918247760031, −5.17286657113310957710550265519, −4.00076781437731183094555519706, −2.06527899559925713482951959891, −0.28103385463971712644577544056, 1.42080812131009881973463061497, 3.53657709274530507466914471242, 4.61810601582922881320620178811, 6.17136497418150592459263975455, 6.56496827695519625730039770967, 7.81128241183804573197548043436, 9.239448377427816619426378828322, 10.43811623390807639719689257908, 10.67237196028687903818419985518, 11.89827128539207862028713176892

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