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Properties

Label 2-252-28.19-c1-0-13
Degree $2$
Conductor $252$
Sign $0.895 + 0.444i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (2.44 + 1.41i)5-s + (0.5 + 2.59i)7-s − 2.82i·8-s + 4·10-s + (−4.89 + 2.82i)11-s − 5.19i·13-s + (2.44 + 2.82i)14-s + (−2.00 − 3.46i)16-s + (−2.44 + 1.41i)17-s + (2.5 − 4.33i)19-s + (4.89 − 2.82i)20-s + (−3.99 + 6.92i)22-s + (−2.44 − 1.41i)23-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + (1.09 + 0.632i)5-s + (0.188 + 0.981i)7-s − 0.999i·8-s + 1.26·10-s + (−1.47 + 0.852i)11-s − 1.44i·13-s + (0.654 + 0.755i)14-s + (−0.500 − 0.866i)16-s + (−0.594 + 0.342i)17-s + (0.573 − 0.993i)19-s + (1.09 − 0.632i)20-s + (−0.852 + 1.47i)22-s + (−0.510 − 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12986 - 0.498945i\)
\(L(\frac12)\) \(\approx\) \(2.12986 - 0.498945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-2.44 - 1.41i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.89 - 2.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (2.44 - 1.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.44 + 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 5.19iT - 43T^{2} \)
47 \( 1 + (-2.44 + 4.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 4.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (-9.79 - 5.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.15431875586780256969894594125, −10.93937066339839870383822637633, −10.27583441028882472697852574264, −9.515908653289793521522510903335, −7.964669319577384183043744636852, −6.60361457132136596147772661406, −5.59583210540170755716544950356, −4.94391502063436488693379766407, −2.88612934852103026029688503894, −2.23923791318654368970529960781, 2.07424347127167627132014105087, 3.80241602635471355342087179505, 5.03638635347303934260229228262, 5.81462826645787011471499938356, 6.99773391736015001621557979536, 8.014557099205812013206611133096, 9.107498577879820666490925181275, 10.32623207637976953258306139051, 11.26431232193847601276473073465, 12.38200521092354997804411692578

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