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Properties

Label 2-1450-29.28-c1-0-20
Degree $2$
Conductor $1450$
Sign $0.999 + 0.0160i$
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 + 0.272i·3-s − 4-s + 0.272·6-s + 4.19·7-s + i·8-s + 2.92·9-s + 2.78i·11-s − 0.272i·12-s + 2.50·13-s − 4.19i·14-s + 16-s + 7.15i·17-s − 2.92i·18-s − 2.27i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.157i·3-s − 0.5·4-s + 0.111·6-s + 1.58·7-s + 0.353i·8-s + 0.975·9-s + 0.838i·11-s − 0.0786i·12-s + 0.695·13-s − 1.12i·14-s + 0.250·16-s + 1.73i·17-s − 0.689i·18-s − 0.521i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.999 + 0.0160i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.999 + 0.0160i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041626390\)
\(L(\frac12)\) \(\approx\) \(2.041626390\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0862 - 5.38i)T \)
good3 \( 1 - 0.272iT - 3T^{2} \)
7 \( 1 - 4.19T + 7T^{2} \)
11 \( 1 - 2.78iT - 11T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 - 7.15iT - 17T^{2} \)
19 \( 1 + 2.27iT - 19T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
31 \( 1 - 0.925iT - 31T^{2} \)
37 \( 1 - 5.64iT - 37T^{2} \)
41 \( 1 + 1.96iT - 41T^{2} \)
43 \( 1 - 8.39iT - 43T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + 9.82T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 9.98iT - 61T^{2} \)
67 \( 1 - 1.74T + 67T^{2} \)
71 \( 1 - 7.21T + 71T^{2} \)
73 \( 1 - 6.43iT - 73T^{2} \)
79 \( 1 + 4.23iT - 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−9.834488365956030556560887305613, −8.558525378761692413681409110975, −8.206257366250072803025410804697, −7.25605112087561648097274667625, −6.15757926667482709135287685378, −5.00906633521172356769452747712, −4.37596791479120607354749490114, −3.65141734088772216244631588751, −1.91727439257532332516288288615, −1.52031308253517073847634249202, 0.934045915260708936989074799316, 2.16371924661641938320783140135, 3.82195250824128874425252070263, 4.53961300027868554834213695288, 5.46127578882881780283662789809, 6.20162755056758319659792780143, 7.30598667136621930137575025318, 7.86715801320662500440625232288, 8.441018463201094337242751239389, 9.397641348162760714007546658863

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