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Properties

Label 2-1450-5.4-c1-0-23
Degree $2$
Conductor $1450$
Sign $-0.447 + 0.894i$
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 − 2.34i·3-s − 4-s − 2.34·6-s + 3.83i·7-s + i·8-s − 2.48·9-s + 3.19·11-s + 2.34i·12-s + 1.14i·13-s + 3.83·14-s + 16-s − 0.803i·17-s + 2.48i·18-s + 2.34·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.35i·3-s − 0.5·4-s − 0.956·6-s + 1.44i·7-s + 0.353i·8-s − 0.829·9-s + 0.963·11-s + 0.676i·12-s + 0.317i·13-s + 1.02·14-s + 0.250·16-s − 0.194i·17-s + 0.586i·18-s + 0.537·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 1450,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.669958388\)
\(L(\frac12)\) \(\approx\) \(1.669958388\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.34iT - 3T^{2} \)
7 \( 1 - 3.83iT - 7T^{2} \)
11 \( 1 - 3.19T + 11T^{2} \)
13 \( 1 - 1.14iT - 13T^{2} \)
17 \( 1 + 0.803iT - 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
23 \( 1 + 8.12iT - 23T^{2} \)
31 \( 1 - 1.90T + 31T^{2} \)
37 \( 1 + 7.63iT - 37T^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 - 7.66iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 0.560iT - 53T^{2} \)
59 \( 1 + 2.36T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 5.12iT - 67T^{2} \)
71 \( 1 - 8.12T + 71T^{2} \)
73 \( 1 - 5.71iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 3.09iT - 83T^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−9.102010881575976026410622867770, −8.612686480759460509980583655270, −7.74049873953026958676343531174, −6.71662946293510979331420534642, −6.12414606854524912966278591941, −5.16377279002638522520865454163, −3.97116776640865590787267829501, −2.61555166663799235648672990572, −2.07318508979689700033427751340, −0.853153442287446562542221416101, 1.12722578318187148976148965090, 3.39832252943560341707242469156, 3.91777118010830431631033251719, 4.66807970390785576666261146898, 5.52801415393760302073098110713, 6.55288882369241774600567765463, 7.39323049263240771129071739424, 8.082906184210834300022284487000, 9.340343168049685012798971203618, 9.536985833939634850437856359605

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