Properties

Label 2-252-63.20-c1-0-2
Degree $2$
Conductor $252$
Sign $0.595 - 0.803i$
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  + (1.10 + 1.33i)3-s + (0.266 + 0.462i)5-s + (2.54 + 0.715i)7-s + (−0.565 + 2.94i)9-s + (−3.39 − 1.96i)11-s + (−0.116 + 0.0674i)13-s + (−0.322 + 0.866i)15-s + 4.32·17-s + 2.22i·19-s + (1.85 + 4.19i)21-s + (−1.70 + 0.983i)23-s + (2.35 − 4.08i)25-s + (−4.55 + 2.49i)27-s + (−5.16 − 2.98i)29-s + (−0.800 + 0.462i)31-s + ⋯
L(s)  = 1  + (0.637 + 0.770i)3-s + (0.119 + 0.206i)5-s + (0.962 + 0.270i)7-s + (−0.188 + 0.982i)9-s + (−1.02 − 0.591i)11-s + (−0.0324 + 0.0187i)13-s + (−0.0832 + 0.223i)15-s + 1.04·17-s + 0.511i·19-s + (0.404 + 0.914i)21-s + (−0.355 + 0.205i)23-s + (0.471 − 0.816i)25-s + (−0.877 + 0.480i)27-s + (−0.959 − 0.553i)29-s + (−0.143 + 0.0829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.595 - 0.803i$
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: \((2,\ 252,\ (\ :1/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42257 + 0.716253i\)
\(L(\frac12)\) \(\approx\) \(1.42257 + 0.716253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 - 1.33i)T \)
7 \( 1 + (-2.54 - 0.715i)T \)
good5 \( 1 + (-0.266 - 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.39 + 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.116 - 0.0674i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + (1.70 - 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.16 + 2.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.800 - 0.462i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (4.59 + 7.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.04 - 5.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + (1.89 + 3.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.35 + 5.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 - 0.381iT - 73T^{2} \)
79 \( 1 + (4.60 - 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.28 - 2.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + (-13.6 - 7.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.07880798573799389176800945532, −10.98353631941174752896308674993, −10.34423367544022023482606898422, −9.324292143331308335186074064259, −8.178389113417505178674508563678, −7.72435107185328648733226608573, −5.80827742551653144000517944814, −4.93417646689399348285905001466, −3.57381425979581670628485448677, −2.25700964571370554301348390799, 1.49932994233771599537261324510, 2.93295995310648152396624011023, 4.59402556642963761441442912456, 5.78779608282861589717564680371, 7.34458824329298176276614294694, 7.78188091838428546558463050183, 8.843525779834986090391648642592, 9.901665559582197268550732951306, 11.03860264091164116229940342951, 12.04684166705411811333497121566

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