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Properties

Label 2-2900-2900.127-c0-0-0
Degree $2$
Conductor $2900$
Sign $0.182 + 0.983i$
Analytic cond. $1.44728$
Root an. cond. $1.20303$
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  + (0.983 − 0.178i)2-s + (0.936 − 0.351i)4-s + (−0.0896 − 0.995i)5-s + (0.858 − 0.512i)8-s + (0.393 − 0.919i)9-s + (−0.266 − 0.963i)10-s + (−0.825 − 0.0556i)13-s + (0.753 − 0.657i)16-s + (0.110 + 0.339i)17-s + (0.222 − 0.974i)18-s + (−0.433 − 0.900i)20-s + (−0.983 + 0.178i)25-s + (−0.821 + 0.0926i)26-s + (−0.919 + 0.393i)29-s + (0.623 − 0.781i)32-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (0.936 − 0.351i)4-s + (−0.0896 − 0.995i)5-s + (0.858 − 0.512i)8-s + (0.393 − 0.919i)9-s + (−0.266 − 0.963i)10-s + (−0.825 − 0.0556i)13-s + (0.753 − 0.657i)16-s + (0.110 + 0.339i)17-s + (0.222 − 0.974i)18-s + (−0.433 − 0.900i)20-s + (−0.983 + 0.178i)25-s + (−0.821 + 0.0926i)26-s + (−0.919 + 0.393i)29-s + (0.623 − 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.279217593\)
\(L(\frac12)\) \(\approx\) \(2.279217593\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.983 + 0.178i)T \)
5 \( 1 + (0.0896 + 0.995i)T \)
29 \( 1 + (0.919 - 0.393i)T \)
good3 \( 1 + (-0.393 + 0.919i)T^{2} \)
7 \( 1 + (-0.974 + 0.222i)T^{2} \)
11 \( 1 + (-0.722 - 0.691i)T^{2} \)
13 \( 1 + (0.825 + 0.0556i)T + (0.990 + 0.134i)T^{2} \)
17 \( 1 + (-0.110 - 0.339i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.919 + 0.393i)T^{2} \)
23 \( 1 + (0.834 - 0.550i)T^{2} \)
31 \( 1 + (0.266 - 0.963i)T^{2} \)
37 \( 1 + (0.0825 + 0.0352i)T + (0.691 + 0.722i)T^{2} \)
41 \( 1 + (-0.306 - 1.93i)T + (-0.951 + 0.309i)T^{2} \)
43 \( 1 + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.473 + 0.880i)T^{2} \)
53 \( 1 + (0.0357 + 1.59i)T + (-0.998 + 0.0448i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.817 - 0.894i)T + (-0.0896 + 0.995i)T^{2} \)
67 \( 1 + (0.880 + 0.473i)T^{2} \)
71 \( 1 + (0.473 + 0.880i)T^{2} \)
73 \( 1 + (0.127 - 0.236i)T + (-0.550 - 0.834i)T^{2} \)
79 \( 1 + (0.178 - 0.983i)T^{2} \)
83 \( 1 + (0.919 - 0.393i)T^{2} \)
89 \( 1 + (-1.64 - 1.13i)T + (0.351 + 0.936i)T^{2} \)
97 \( 1 + (0.186 + 0.498i)T + (-0.753 + 0.657i)T^{2} \)
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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

−8.835106515510831796491553483696, −7.88366763451889042590379431093, −7.16056782302490360884395190065, −6.34851307122872653618547843631, −5.54784076446559007961357481368, −4.84417651451848513488130516061, −4.09429014202109669754369352310, −3.36941251913407480421990450667, −2.15614365289140111840512828152, −1.08025555410597962431276253592, 2.00734545385170837113413840796, 2.60356950210803635201273234311, 3.62663296805478063766515573441, 4.40791683075017972697232851454, 5.28203309782983672467994194650, 5.95716368938004215220998878455, 6.94687458779665223237522398417, 7.41865676057712328124203706831, 7.901306869371111226541856757587, 9.126719451320511941984685044413

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