Properties

Label 2-252-63.41-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.424 - 0.905i$
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  + (0.680 + 1.59i)3-s + (−2.09 + 3.62i)5-s + (1.36 − 2.26i)7-s + (−2.07 + 2.16i)9-s + (−1.23 + 0.711i)11-s + (−0.850 − 0.491i)13-s + (−7.19 − 0.866i)15-s + 0.370·17-s + 4.97i·19-s + (4.53 + 0.638i)21-s + (4.98 + 2.87i)23-s + (−6.26 − 10.8i)25-s + (−4.86 − 1.82i)27-s + (7.31 − 4.22i)29-s + (6.28 + 3.62i)31-s + ⋯
L(s)  = 1  + (0.392 + 0.919i)3-s + (−0.936 + 1.62i)5-s + (0.517 − 0.855i)7-s + (−0.691 + 0.722i)9-s + (−0.371 + 0.214i)11-s + (−0.235 − 0.136i)13-s + (−1.85 − 0.223i)15-s + 0.0899·17-s + 1.14i·19-s + (0.990 + 0.139i)21-s + (1.04 + 0.600i)23-s + (−1.25 − 2.17i)25-s + (−0.936 − 0.351i)27-s + (1.35 − 0.784i)29-s + (1.12 + 0.651i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621091 + 0.977357i\)
\(L(\frac12)\) \(\approx\) \(0.621091 + 0.977357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.680 - 1.59i)T \)
7 \( 1 + (-1.36 + 2.26i)T \)
good5 \( 1 + (2.09 - 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.23 - 0.711i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.850 + 0.491i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.370T + 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 + (-4.98 - 2.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.31 + 4.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.28 - 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + (1.06 - 1.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.13 + 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.97iT - 53T^{2} \)
59 \( 1 + (2.27 - 3.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.03 + 8.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 + (-4.25 - 7.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.972 - 1.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 + (3.34 - 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.99274303992492431925630169688, −11.12732249305481337328041687265, −10.46813032887237464241087472234, −9.868227194670124547606177170348, −8.150761812281379023942669431583, −7.66136265022230891495544728769, −6.49983978707374764275746521486, −4.79417545136496069286922167970, −3.75332600022739027556260021412, −2.80936715308775075091279209950, 0.939931289483233095480732753721, 2.70636470832980959822848747470, 4.52351518714157562289266144069, 5.42010449303368959634858363420, 6.93491387226775733930167708827, 8.156819431206871779945142888600, 8.531576910593888228923915205587, 9.344828387238397442754524373604, 11.25793584708318856006118363249, 11.93299092411639476988631704432

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