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Properties

Label 2-252-252.223-c1-0-8
Degree $2$
Conductor $252$
Sign $0.308 - 0.951i$
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  + (0.325 − 1.37i)2-s + (0.844 + 1.51i)3-s + (−1.78 − 0.895i)4-s + (−3.45 + 1.99i)5-s + (2.35 − 0.670i)6-s + (−0.447 + 2.60i)7-s + (−1.81 + 2.17i)8-s + (−1.57 + 2.55i)9-s + (1.62 + 5.40i)10-s + (1.72 + 0.997i)11-s + (−0.156 − 3.46i)12-s + (2.38 − 1.37i)13-s + (3.44 + 1.46i)14-s + (−5.93 − 3.53i)15-s + (2.39 + 3.20i)16-s − 2.88i·17-s + ⋯
L(s)  = 1  + (0.229 − 0.973i)2-s + (0.487 + 0.873i)3-s + (−0.894 − 0.447i)4-s + (−1.54 + 0.891i)5-s + (0.961 − 0.273i)6-s + (−0.169 + 0.985i)7-s + (−0.641 + 0.767i)8-s + (−0.524 + 0.851i)9-s + (0.512 + 1.70i)10-s + (0.521 + 0.300i)11-s + (−0.0451 − 0.998i)12-s + (0.661 − 0.381i)13-s + (0.920 + 0.391i)14-s + (−1.53 − 0.913i)15-s + (0.599 + 0.800i)16-s − 0.700i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.308 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773854 + 0.562501i\)
\(L(\frac12)\) \(\approx\) \(0.773854 + 0.562501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.325 + 1.37i)T \)
3 \( 1 + (-0.844 - 1.51i)T \)
7 \( 1 + (0.447 - 2.60i)T \)
good5 \( 1 + (3.45 - 1.99i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.72 - 0.997i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.38 + 1.37i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.88iT - 17T^{2} \)
19 \( 1 - 0.0855T + 19T^{2} \)
23 \( 1 + (5.06 - 2.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 - 4.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.893 - 1.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.22T + 37T^{2} \)
41 \( 1 + (-3.94 + 2.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.28 - 5.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.22 - 3.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.30T + 53T^{2} \)
59 \( 1 + (-2.41 - 4.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.08 - 4.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.76 - 1.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.57iT - 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + (11.4 + 6.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.56 + 14.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.76iT - 89T^{2} \)
97 \( 1 + (4.07 + 2.35i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.82699638085285383856432648627, −11.39784068258488032796351173593, −10.51781952518196317434501046140, −9.469705665275568896357453416844, −8.612867750047309868426774811856, −7.62869735562129396491180063334, −5.85997849954321266095983726913, −4.43334313936915455109725580244, −3.55834833711146897524284863755, −2.68019963861532879158001100273, 0.70003321708352232971915731833, 3.77651154426932134100730127650, 4.18854661816324728557521022349, 6.04131523731908277787714800131, 7.04971375783471896641737109423, 7.974601670183983642992393809893, 8.381962565611271990906412137436, 9.440569820329498200924284231742, 11.25008520632334687049400791327, 12.18980289274979579345890219038

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