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Properties

Label 2-252-9.4-c1-0-2
Degree $2$
Conductor $252$
Sign $0.927 - 0.373i$
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  + (−1.71 − 0.272i)3-s + (−0.119 + 0.207i)5-s + (0.5 + 0.866i)7-s + (2.85 + 0.931i)9-s + (2.56 + 4.43i)11-s + (2.44 − 4.23i)13-s + (0.260 − 0.321i)15-s + 3.70·17-s + 3.66·19-s + (−0.619 − 1.61i)21-s + (−3.71 + 6.42i)23-s + (2.47 + 4.28i)25-s + (−4.62 − 2.36i)27-s + (−1.73 − 3.00i)29-s + (0.358 − 0.621i)31-s + ⋯
L(s)  = 1  + (−0.987 − 0.157i)3-s + (−0.0534 + 0.0926i)5-s + (0.188 + 0.327i)7-s + (0.950 + 0.310i)9-s + (0.772 + 1.33i)11-s + (0.677 − 1.17i)13-s + (0.0673 − 0.0830i)15-s + 0.898·17-s + 0.839·19-s + (−0.135 − 0.352i)21-s + (−0.773 + 1.34i)23-s + (0.494 + 0.856i)25-s + (−0.890 − 0.455i)27-s + (−0.321 − 0.557i)29-s + (0.0644 − 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987593 + 0.191240i\)
\(L(\frac12)\) \(\approx\) \(0.987593 + 0.191240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.272i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.119 - 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.56 - 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + (3.71 - 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.358 + 0.621i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.16 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.942T + 53T^{2} \)
59 \( 1 + (3.78 - 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.75 + 4.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.330 + 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 + (-3.11 - 5.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.85 + 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + (-8.57 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.97799351870408510048550690269, −11.37693140805465257400742687260, −10.17715003106063684072318641559, −9.527149679089405759578863469670, −7.930476536145668280255029103203, −7.12971908713605981445671675979, −5.88775049970322527640907444623, −5.09276305379219344796297682095, −3.63719118170593401263908072317, −1.49537085468508805979606334047, 1.13148452339232045201304986063, 3.60962090874277268982833436381, 4.69991105010304886346018262146, 6.02837776730201479444681698013, 6.67220598287954726968658860231, 8.090643612147532324380121976392, 9.178984024196770397496498408289, 10.25084022203673387344958845302, 11.22211114282473288925936106151, 11.74350293369536384169631102504

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