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Properties

Label 2-2900-2900.147-c0-0-0
Degree $2$
Conductor $2900$
Sign $0.997 - 0.0672i$
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  + (0.936 − 0.351i)2-s + (0.753 − 0.657i)4-s + (0.178 + 0.983i)5-s + (0.473 − 0.880i)8-s + (0.691 + 0.722i)9-s + (0.512 + 0.858i)10-s + (−0.107 + 0.0815i)13-s + (0.134 − 0.990i)16-s + (−0.568 + 0.413i)17-s + (0.900 + 0.433i)18-s + (0.781 + 0.623i)20-s + (−0.936 + 0.351i)25-s + (−0.0715 + 0.113i)26-s + (0.722 + 0.691i)29-s + (−0.222 − 0.974i)32-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (0.753 − 0.657i)4-s + (0.178 + 0.983i)5-s + (0.473 − 0.880i)8-s + (0.691 + 0.722i)9-s + (0.512 + 0.858i)10-s + (−0.107 + 0.0815i)13-s + (0.134 − 0.990i)16-s + (−0.568 + 0.413i)17-s + (0.900 + 0.433i)18-s + (0.781 + 0.623i)20-s + (−0.936 + 0.351i)25-s + (−0.0715 + 0.113i)26-s + (0.722 + 0.691i)29-s + (−0.222 − 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.997 - 0.0672i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ 0.997 - 0.0672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.379884921\)
\(L(\frac12)\) \(\approx\) \(2.379884921\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.936 + 0.351i)T \)
5 \( 1 + (-0.178 - 0.983i)T \)
29 \( 1 + (-0.722 - 0.691i)T \)
good3 \( 1 + (-0.691 - 0.722i)T^{2} \)
7 \( 1 + (0.433 + 0.900i)T^{2} \)
11 \( 1 + (0.998 - 0.0448i)T^{2} \)
13 \( 1 + (0.107 - 0.0815i)T + (0.266 - 0.963i)T^{2} \)
17 \( 1 + (0.568 - 0.413i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.722 + 0.691i)T^{2} \)
23 \( 1 + (-0.919 - 0.393i)T^{2} \)
31 \( 1 + (-0.512 + 0.858i)T^{2} \)
37 \( 1 + (-1.43 + 1.37i)T + (0.0448 - 0.998i)T^{2} \)
41 \( 1 + (0.338 - 0.663i)T + (-0.587 - 0.809i)T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.550 + 0.834i)T^{2} \)
53 \( 1 + (0.728 + 0.665i)T + (0.0896 + 0.995i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.33 - 1.11i)T + (0.178 - 0.983i)T^{2} \)
67 \( 1 + (0.834 - 0.550i)T^{2} \)
71 \( 1 + (-0.550 + 0.834i)T^{2} \)
73 \( 1 + (1.06 + 1.60i)T + (-0.393 + 0.919i)T^{2} \)
79 \( 1 + (0.351 - 0.936i)T^{2} \)
83 \( 1 + (-0.722 - 0.691i)T^{2} \)
89 \( 1 + (-1.44 - 0.658i)T + (0.657 + 0.753i)T^{2} \)
97 \( 1 + (-0.674 - 0.772i)T + (-0.134 + 0.990i)T^{2} \)
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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

−9.154795185943927387229535860127, −7.901749335225048005671588832429, −7.27591481759707107155584375361, −6.55056275996373541663695823352, −5.94080510366508246439652605074, −4.93269422581849564136231982240, −4.25574918981306275644496984870, −3.32655939173015874900573430015, −2.44883433225657572386641969143, −1.64893567106713660218111733317, 1.27363138813514709496863086402, 2.47763672912042661082021202004, 3.53976933143065026200354701663, 4.56534182526508060734595471038, 4.75534937229926840765013562279, 5.98737117914127526382184151299, 6.40180054276387697111025933072, 7.37618044627894546742702533870, 8.041700670346038614136636447778, 8.867503125381152057138467398375

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