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Properties

Label 2-252-63.20-c1-0-3
Degree $2$
Conductor $252$
Sign $0.925 - 0.378i$
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  + (1.72 + 0.136i)3-s + (0.276 + 0.479i)5-s + (−1.98 + 1.74i)7-s + (2.96 + 0.472i)9-s + (4.03 + 2.32i)11-s + (3.58 − 2.06i)13-s + (0.412 + 0.866i)15-s − 7.24·17-s − 6.71i·19-s + (−3.67 + 2.74i)21-s + (−4.85 + 2.80i)23-s + (2.34 − 4.06i)25-s + (5.05 + 1.22i)27-s + (1.16 + 0.673i)29-s + (−0.830 + 0.479i)31-s + ⋯
L(s)  = 1  + (0.996 + 0.0789i)3-s + (0.123 + 0.214i)5-s + (−0.751 + 0.660i)7-s + (0.987 + 0.157i)9-s + (1.21 + 0.702i)11-s + (0.993 − 0.573i)13-s + (0.106 + 0.223i)15-s − 1.75·17-s − 1.54i·19-s + (−0.800 + 0.598i)21-s + (−1.01 + 0.584i)23-s + (0.469 − 0.812i)25-s + (0.972 + 0.234i)27-s + (0.216 + 0.125i)29-s + (−0.149 + 0.0861i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66130 + 0.326784i\)
\(L(\frac12)\) \(\approx\) \(1.66130 + 0.326784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.136i)T \)
7 \( 1 + (1.98 - 1.74i)T \)
good5 \( 1 + (-0.276 - 0.479i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.03 - 2.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.58 + 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.24T + 17T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.16 - 0.673i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 - 0.479i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.90 - 8.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.43iT - 53T^{2} \)
59 \( 1 + (-3.89 - 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.37 + 3.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.407iT - 71T^{2} \)
73 \( 1 + 8.63iT - 73T^{2} \)
79 \( 1 + (0.318 - 0.551i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.78 + 4.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.93T + 89T^{2} \)
97 \( 1 + (-7.48 - 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.25928090258939121550780371785, −11.09824924880751772561035611188, −9.961643176357680443138408388936, −9.056158173544166829596455282252, −8.571743505207234594852517762416, −7.00442068822172946995559465461, −6.34843099343579247237129816692, −4.56070146778665769132017159807, −3.37504205564567990401870780498, −2.10689603886908677230204034580, 1.64437967367432163672492386897, 3.51950247561280760767113132114, 4.17056046957023460316070147961, 6.25211943119332338844754433348, 6.89550516159051552712025928622, 8.369822654087735270295906647400, 8.951797493461574824220963623335, 9.899283393484934326467865105911, 10.93367658565812057543937286927, 12.11606781649245744078451637736

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