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Properties

Label 2-2900-29.28-c1-0-44
Degree $2$
Conductor $2900$
Sign $0.185 + 0.982i$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
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  + 2·7-s + 3·9-s − 5.29i·11-s + 2·13-s − 5.29i·19-s − 6·23-s + (−1 − 5.29i)29-s − 5.29i·31-s + 10.5i·43-s − 10.5i·47-s − 3·49-s + 2·53-s + 10.5i·61-s + 6·63-s − 10·67-s + ⋯
L(s)  = 1  + 0.755·7-s + 9-s − 1.59i·11-s + 0.554·13-s − 1.21i·19-s − 1.25·23-s + (−0.185 − 0.982i)29-s − 0.950i·31-s + 1.61i·43-s − 1.54i·47-s − 0.428·49-s + 0.274·53-s + 1.35i·61-s + 0.755·63-s − 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.185 + 0.982i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 0.185 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.988394511\)
\(L(\frac12)\) \(\approx\) \(1.988394511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 + (1 + 5.29i)T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 5.29iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−8.470582301378564517997574557205, −7.976562544665603274110148197718, −7.15512741440067962975235820955, −6.20752930886461033362313003889, −5.65102091194312059907869035499, −4.54963178960577385454365137426, −3.96779126658561315776452495989, −2.89324064508001135448263985147, −1.75233454512257633143037836041, −0.64375026641997947079017916041, 1.51907464393495957692269158149, 1.92863160573567488285900611370, 3.48747735092341229806242415675, 4.31884821915215684365169176836, 4.88896816840571907954329386207, 5.87192877836553894179469530398, 6.79512126214371596950172790857, 7.48649479897347423833606477601, 8.027080642777572456683984381219, 8.935178472768283497828851008052

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