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Properties

Label 2-252-12.11-c1-0-2
Degree $2$
Conductor $252$
Sign $-0.399 - 0.916i$
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  + (0.892 + 1.09i)2-s + (−0.406 + 1.95i)4-s + 2.56i·5-s − i·7-s + (−2.51 + 1.30i)8-s + (−2.81 + 2.28i)10-s + 1.15·11-s − 0.578·13-s + (1.09 − 0.892i)14-s + (−3.66 − 1.59i)16-s + 5.39i·17-s − 6.20i·19-s + (−5.02 − 1.04i)20-s + (1.02 + 1.26i)22-s + 7.62·23-s + ⋯
L(s)  = 1  + (0.631 + 0.775i)2-s + (−0.203 + 0.979i)4-s + 1.14i·5-s − 0.377i·7-s + (−0.887 + 0.460i)8-s + (−0.889 + 0.723i)10-s + 0.346·11-s − 0.160·13-s + (0.293 − 0.238i)14-s + (−0.917 − 0.398i)16-s + 1.30i·17-s − 1.42i·19-s + (−1.12 − 0.233i)20-s + (0.218 + 0.269i)22-s + 1.58·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889648 + 1.35767i\)
\(L(\frac12)\) \(\approx\) \(0.889648 + 1.35767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.892 - 1.09i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 2.56iT - 5T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 + 0.578T + 13T^{2} \)
17 \( 1 - 5.39iT - 17T^{2} \)
19 \( 1 + 6.20iT - 19T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 5.04iT - 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 + 6.21iT - 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 4.53iT - 53T^{2} \)
59 \( 1 + 4.83T + 59T^{2} \)
61 \( 1 - 0.951T + 61T^{2} \)
67 \( 1 - 2.78iT - 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 - 5.68iT - 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−12.59936770324829916575114964142, −11.34001004809531703315736713909, −10.69244954383667486861969152729, −9.306950392581247083498355070637, −8.207162859038336147471730871651, −6.96505823240193969429507603336, −6.64141264006854715666951807654, −5.21877230465458612809107258779, −3.92021377274387318283701870339, −2.79124229711119374780670335298, 1.23072839504430405177736313609, 2.94586883651397238840191389276, 4.46168541935701723252020376283, 5.22299617949174260747743584649, 6.36394940234821232285647028116, 7.988288373386735556965408636606, 9.252625049378327277979003678428, 9.658615130994367332749548982539, 11.12969610435797055772539856415, 11.82515944626247818829367756768

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