Properties

Label 2-1450-5.4-c1-0-19
Degree $2$
Conductor $1450$
Sign $0.894 - 0.447i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  + i·2-s − 1.30i·3-s − 4-s + 1.30·6-s + 0.302i·7-s i·8-s + 1.30·9-s + 4.60·11-s + 1.30i·12-s + 1.30i·13-s − 0.302·14-s + 16-s + 2.30i·17-s + 1.30i·18-s − 6.60·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.752i·3-s − 0.5·4-s + 0.531·6-s + 0.114i·7-s − 0.353i·8-s + 0.434·9-s + 1.38·11-s + 0.376i·12-s + 0.361i·13-s − 0.0809·14-s + 0.250·16-s + 0.558i·17-s + 0.307i·18-s − 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784016725\)
\(L(\frac12)\) \(\approx\) \(1.784016725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + 1.30iT - 3T^{2} \)
7 \( 1 - 0.302iT - 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 - 1.30iT - 13T^{2} \)
17 \( 1 - 2.30iT - 17T^{2} \)
19 \( 1 + 6.60T + 19T^{2} \)
23 \( 1 - 3.90iT - 23T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 6.60iT - 37T^{2} \)
41 \( 1 - 1.39T + 41T^{2} \)
43 \( 1 + 0.302iT - 43T^{2} \)
47 \( 1 - 9.21iT - 47T^{2} \)
53 \( 1 + 9.69iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 3.30T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 1.90T + 79T^{2} \)
83 \( 1 - 7.81iT - 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 9.11iT - 97T^{2} \)
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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

−9.407505941436812951230021169120, −8.585120469202120106853607938620, −7.952801899701666391133925083357, −6.89758752111149639965100540499, −6.56772075218612425649020988405, −5.77685282411163153341699446803, −4.42803433230484454317659436846, −3.88008541843714420732471338040, −2.19773890167835570569176934007, −1.07662233890556900058323718633, 0.968309749179320741965891951673, 2.33639587063362683445267341666, 3.52855398452850451515124902677, 4.31935601468584348403480455592, 4.84119530760227908915465267476, 6.21772236497891081578530631090, 6.89600837710128038279716916555, 8.174594712650930415695353148814, 8.896228826786564593252610850945, 9.573951191825616422015654732021

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