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Properties

Label 2-252-63.5-c1-0-5
Degree $2$
Conductor $252$
Sign $-0.714 + 0.699i$
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 + 1.06i)3-s + (−1.48 − 2.57i)5-s + (−0.200 + 2.63i)7-s + (0.727 − 2.91i)9-s + (−4.09 − 2.36i)11-s + (−3.54 − 2.04i)13-s + (4.76 + 1.92i)15-s + (−0.835 − 1.44i)17-s + (−4.25 − 2.45i)19-s + (−2.53 − 3.81i)21-s + (4.25 − 2.45i)23-s + (−1.91 + 3.30i)25-s + (2.11 + 4.74i)27-s + (0.238 − 0.137i)29-s + 1.60i·31-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)3-s + (−0.664 − 1.15i)5-s + (−0.0756 + 0.997i)7-s + (0.242 − 0.970i)9-s + (−1.23 − 0.712i)11-s + (−0.981 − 0.566i)13-s + (1.23 + 0.497i)15-s + (−0.202 − 0.350i)17-s + (−0.975 − 0.563i)19-s + (−0.554 − 0.832i)21-s + (0.886 − 0.511i)23-s + (−0.382 + 0.661i)25-s + (0.406 + 0.913i)27-s + (0.0442 − 0.0255i)29-s + 0.287i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105510 - 0.258730i\)
\(L(\frac12)\) \(\approx\) \(0.105510 - 0.258730i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 1.06i)T \)
7 \( 1 + (0.200 - 2.63i)T \)
good5 \( 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.54 + 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.835 + 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.25 + 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.25 + 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.238 + 0.137i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.60iT - 31T^{2} \)
37 \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.55 - 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (-0.707 + 0.408i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 7.20iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 + 7.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (4.03 + 6.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.60 + 7.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.00 + 4.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.66724447514650471598200507538, −10.89262545333387802450800948817, −9.757377079119598948694572869022, −8.778280651712349969808890173945, −8.009393477666694280562436734141, −6.40824922356328489033978721404, −5.07964929136928687443205564004, −4.82620113995337113197257722943, −2.95998324559394378168046820525, −0.22864093968986012850567952338, 2.30246978504614642179098247346, 4.01011638260010772254559842263, 5.25610276045664068180257969860, 6.85420150946954883586381214832, 7.15642536285065620422654566690, 8.060208594711258565273431907712, 10.00768633502968982450866964142, 10.64754555916667061177544533536, 11.30255371687315149885976794928, 12.40174488918383262118011062148

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