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Properties

Label 2-252-252.223-c1-0-33
Degree $2$
Conductor $252$
Sign $0.513 + 0.858i$
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  + (1.36 − 0.366i)2-s − 1.73·3-s + (1.73 − i)4-s + (1.5 − 0.866i)5-s + (−2.36 + 0.633i)6-s + (−1.73 − 2i)7-s + (1.99 − 2i)8-s + 2.99·9-s + (1.73 − 1.73i)10-s + (0.866 + 0.5i)11-s + (−2.99 + 1.73i)12-s + (−1.5 + 0.866i)13-s + (−3.09 − 2.09i)14-s + (−2.59 + 1.49i)15-s + (1.99 − 3.46i)16-s − 3.46i·17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s − 1.00·3-s + (0.866 − 0.5i)4-s + (0.670 − 0.387i)5-s + (−0.965 + 0.258i)6-s + (−0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (0.547 − 0.547i)10-s + (0.261 + 0.150i)11-s + (−0.866 + 0.499i)12-s + (−0.416 + 0.240i)13-s + (−0.827 − 0.560i)14-s + (−0.670 + 0.387i)15-s + (0.499 − 0.866i)16-s − 0.840i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.513 + 0.858i$
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: \((2,\ 252,\ (\ :1/2),\ 0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52391 - 0.864060i\)
\(L(\frac12)\) \(\approx\) \(1.52391 - 0.864060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 + 1.73T \)
7 \( 1 + (1.73 + 2i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + (4.33 - 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.33 - 7.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (7.5 - 4.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.59 - 4.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.9 + 7.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (2.59 + 1.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.866 + 1.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.3iT - 89T^{2} \)
97 \( 1 + (-4.5 - 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.95932691798424940933058372924, −11.21912911321892133451407519481, −9.949325310176182142432143922828, −9.686893887253336176687193247729, −7.35983071920907917120746718290, −6.66933734433112663769341375752, −5.53566865922893692989759711715, −4.78691729555622263933618974886, −3.41211065654564407845841869545, −1.38502692382542142657589191908, 2.28882399713801770003505012401, 3.84068279230267176640335174045, 5.31908346654087518336461275528, 6.00379419664960158252055012288, 6.69402091778525476337320543773, 7.960929673871615105542199046822, 9.687398935950693204645148670569, 10.38062255326511808218598267002, 11.70024553577640956998334993896, 12.07509154099649335590872545571

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