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Properties

Label 2-1452-33.26-c0-0-5
Degree $2$
Conductor $1452$
Sign $-0.642 + 0.766i$
Analytic cond. $0.724642$
Root an. cond. $0.851259$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (−1.61 − 1.17i)13-s + (−0.309 + 0.951i)19-s − 0.999·21-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (0.809 + 0.587i)31-s + (−0.309 − 0.951i)37-s + (−1.61 + 1.17i)39-s + 2·43-s + (0.809 + 0.587i)57-s + (0.809 − 0.587i)61-s + (−0.309 + 0.951i)63-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (−1.61 − 1.17i)13-s + (−0.309 + 0.951i)19-s − 0.999·21-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (0.809 + 0.587i)31-s + (−0.309 − 0.951i)37-s + (−1.61 + 1.17i)39-s + 2·43-s + (0.809 + 0.587i)57-s + (0.809 − 0.587i)61-s + (−0.309 + 0.951i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(0.724642\)
Root analytic conductor: \(0.851259\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9284100275\)
\(L(\frac12)\) \(\approx\) \(0.9284100275\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.451369085110863188575882756914, −8.395077936672841858858736916609, −7.68548686572895365515475185009, −7.19170022028651717731064058856, −6.30716183709795566926610724160, −5.40072237263758534397085337937, −4.24481931812450309555751200454, −3.14207717079243753562847926677, −2.24444897088159940053616511441, −0.69952496655136278319894867673, 2.28336631182037581080680644671, 2.88948847286368683400304257951, 4.22582064560975869019590040608, 4.87766048960526334035773386020, 5.72239156696724876445176673654, 6.77091505978560696282802040059, 7.64964874028135589436799757753, 8.755659378192548779166291219695, 9.246835881461318603233335283477, 9.757520770243262478065007699644

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