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Properties

Label 2-252-63.58-c1-0-5
Degree $2$
Conductor $252$
Sign $0.0571 + 0.998i$
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  + (−0.492 + 1.66i)3-s + (−1.80 − 3.12i)5-s + (−1.02 − 2.44i)7-s + (−2.51 − 1.63i)9-s + (3.01 − 5.21i)11-s + (−2.55 + 4.42i)13-s + (6.06 − 1.45i)15-s + (0.111 + 0.192i)17-s + (1.71 − 2.96i)19-s + (4.55 − 0.492i)21-s + (0.509 + 0.883i)23-s + (−3.99 + 6.91i)25-s + (3.95 − 3.36i)27-s + (−2.83 − 4.91i)29-s − 5.04·31-s + ⋯
L(s)  = 1  + (−0.284 + 0.958i)3-s + (−0.805 − 1.39i)5-s + (−0.386 − 0.922i)7-s + (−0.838 − 0.545i)9-s + (0.908 − 1.57i)11-s + (−0.709 + 1.22i)13-s + (1.56 − 0.375i)15-s + (0.0269 + 0.0466i)17-s + (0.392 − 0.680i)19-s + (0.994 − 0.107i)21-s + (0.106 + 0.184i)23-s + (−0.798 + 1.38i)25-s + (0.761 − 0.648i)27-s + (−0.526 − 0.912i)29-s − 0.906·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.0571 + 0.998i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.0571 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540640 - 0.510592i\)
\(L(\frac12)\) \(\approx\) \(0.540640 - 0.510592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.492 - 1.66i)T \)
7 \( 1 + (1.02 + 2.44i)T \)
good5 \( 1 + (1.80 + 3.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.01 + 5.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.55 - 4.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.111 - 0.192i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.71 + 2.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.509 - 0.883i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.83 + 4.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.04T + 31T^{2} \)
37 \( 1 + (-1.68 + 2.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0955 + 0.165i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.71 - 2.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.07T + 47T^{2} \)
53 \( 1 + (2.65 + 4.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.59T + 59T^{2} \)
61 \( 1 - 1.78T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 5.89T + 71T^{2} \)
73 \( 1 + (-6.30 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (3.59 + 6.23i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.44 - 7.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.72 + 4.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.45806465053635961129809155536, −11.25352919418005105630949089482, −9.611388990720620615202791448472, −9.129017010592206667201284956652, −8.138451601744527988999412525536, −6.74295223142651013311940890658, −5.39631958420748348123074675803, −4.27827507804680791110917952197, −3.66460829272018783671830403941, −0.61055075501213933325141314681, 2.25159853448712322578282531929, 3.42795247126110740925294418258, 5.29811662324748490775764519898, 6.54755007996178003046881341698, 7.22888597515771543815051010972, 7.994655979124150294479266213339, 9.472576588394131722044658641802, 10.52618623867818617421035756478, 11.54415575861382228292832842861, 12.30207125016125352275672826299

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