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Properties

Label 2-252-252.11-c1-0-23
Degree $2$
Conductor $252$
Sign $0.996 - 0.0828i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 0.599i)2-s + (1.73 + 0.0599i)3-s + (1.28 − 1.53i)4-s + (0.426 + 0.246i)5-s + (−2.25 + 0.960i)6-s + (1.23 − 2.34i)7-s + (−0.723 + 2.73i)8-s + (2.99 + 0.207i)9-s + (−0.694 − 0.0600i)10-s + (−1.12 − 1.94i)11-s + (2.31 − 2.57i)12-s + (0.169 + 0.293i)13-s + (−0.177 + 3.73i)14-s + (0.724 + 0.452i)15-s + (−0.711 − 3.93i)16-s + (0.305 + 0.176i)17-s + ⋯
L(s)  = 1  + (−0.905 + 0.423i)2-s + (0.999 + 0.0345i)3-s + (0.641 − 0.767i)4-s + (0.190 + 0.110i)5-s + (−0.919 + 0.391i)6-s + (0.465 − 0.884i)7-s + (−0.255 + 0.966i)8-s + (0.997 + 0.0691i)9-s + (−0.219 − 0.0189i)10-s + (−0.339 − 0.587i)11-s + (0.667 − 0.744i)12-s + (0.0469 + 0.0813i)13-s + (−0.0473 + 0.998i)14-s + (0.186 + 0.116i)15-s + (−0.177 − 0.984i)16-s + (0.0740 + 0.0427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.996 - 0.0828i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.996 - 0.0828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23861 + 0.0513802i\)
\(L(\frac12)\) \(\approx\) \(1.23861 + 0.0513802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.599i)T \)
3 \( 1 + (-1.73 - 0.0599i)T \)
7 \( 1 + (-1.23 + 2.34i)T \)
good5 \( 1 + (-0.426 - 0.246i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.12 + 1.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.169 - 0.293i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.305 - 0.176i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 + 0.598i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.76 - 4.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.21 - 3.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.16iT - 31T^{2} \)
37 \( 1 + (-1.16 - 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.03 + 1.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.82 - 1.05i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.14T + 47T^{2} \)
53 \( 1 + (8.98 + 5.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.13T + 59T^{2} \)
61 \( 1 + 6.63T + 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + (1.82 - 3.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.75iT - 79T^{2} \)
83 \( 1 + (-8.88 + 15.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.75 + 5.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.93 - 13.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.84218556549190746372089692131, −10.64371756100077233070089757817, −10.09203715617097183032201828656, −9.023257582739368631755259692957, −8.124255283176833142758017933402, −7.45830603173681806096786908739, −6.36359237212390086924068466810, −4.80836609708507078101309895541, −3.15426106456718533034932860127, −1.51442083270512003714974935088, 1.85590222762905175060341264766, 2.85111924645890602815139850854, 4.42501394317360684152793835101, 6.20420274192453315887999751917, 7.65782246076172508513893748614, 8.163415594151634517237865671999, 9.254572452029332505127254951922, 9.786304461899939552490441100696, 10.91995363919962548190221992883, 12.06762605566825068415608091623

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