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Properties

Label 2-1900-95.94-c2-0-54
Degree $2$
Conductor $1900$
Sign $0.0594 + 0.998i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.83·3-s − 3.72i·7-s + 14.3·9-s − 13.8·11-s − 4.10·13-s − 15.3i·17-s + (16.4 − 9.49i)19-s − 18.0i·21-s − 17.4i·23-s + 25.8·27-s − 16.3i·29-s + 15.9i·31-s − 67.1·33-s + 5.14·37-s − 19.8·39-s + ⋯
L(s)  = 1  + 1.61·3-s − 0.532i·7-s + 1.59·9-s − 1.26·11-s − 0.315·13-s − 0.900i·17-s + (0.866 − 0.499i)19-s − 0.857i·21-s − 0.758i·23-s + 0.956·27-s − 0.565i·29-s + 0.513i·31-s − 2.03·33-s + 0.139·37-s − 0.508·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0594 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0594 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.0594 + 0.998i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ 0.0594 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.970310215\)
\(L(\frac12)\) \(\approx\) \(2.970310215\)
\(L(2)\) 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 \( 1 \)
19 \( 1 + (-16.4 + 9.49i)T \)
good3 \( 1 - 4.83T + 9T^{2} \)
7 \( 1 + 3.72iT - 49T^{2} \)
11 \( 1 + 13.8T + 121T^{2} \)
13 \( 1 + 4.10T + 169T^{2} \)
17 \( 1 + 15.3iT - 289T^{2} \)
23 \( 1 + 17.4iT - 529T^{2} \)
29 \( 1 + 16.3iT - 841T^{2} \)
31 \( 1 - 15.9iT - 961T^{2} \)
37 \( 1 - 5.14T + 1.36e3T^{2} \)
41 \( 1 + 42.2iT - 1.68e3T^{2} \)
43 \( 1 + 1.56iT - 1.84e3T^{2} \)
47 \( 1 + 54.5iT - 2.20e3T^{2} \)
53 \( 1 + 55.9T + 2.80e3T^{2} \)
59 \( 1 + 101. iT - 3.48e3T^{2} \)
61 \( 1 - 80.5T + 3.72e3T^{2} \)
67 \( 1 - 24.8T + 4.48e3T^{2} \)
71 \( 1 - 121. iT - 5.04e3T^{2} \)
73 \( 1 - 33.9iT - 5.32e3T^{2} \)
79 \( 1 + 102. iT - 6.24e3T^{2} \)
83 \( 1 + 80.3iT - 6.88e3T^{2} \)
89 \( 1 - 12.0iT - 7.92e3T^{2} \)
97 \( 1 + 161.T + 9.40e3T^{2} \)
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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

−8.725253385845712188236896849258, −8.120299831397545821025576009331, −7.39047560328194971314669789773, −6.91995533199762498894018626491, −5.42249869777253926986144308086, −4.63382140566231457845721486659, −3.60248521050377796027774545388, −2.80305330506275873152709693894, −2.13955207723703141587768060114, −0.57070091718281573031347084476, 1.49579565006452044014839972084, 2.50903128412804267799140849898, 3.11837445219327313268718122841, 4.04037115943788227334840791899, 5.14222319711227363819396970801, 5.97524916625613082161691253166, 7.23631037237928776053050256150, 7.896201298371695917440461113822, 8.288773915954756793570566239129, 9.214051205739752535643185914029

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