Properties

Label 2-1900-95.94-c2-0-5
Degree $2$
Conductor $1900$
Sign $-0.968 + 0.247i$
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  − 2.16·3-s + 8.18i·7-s − 4.29·9-s + 3.30·11-s + 8.86·13-s + 10.2i·17-s + (−18.5 − 4.03i)19-s − 17.7i·21-s − 0.896i·23-s + 28.8·27-s + 17.9i·29-s + 40.8i·31-s − 7.16·33-s − 7.68·37-s − 19.2·39-s + ⋯
L(s)  = 1  − 0.723·3-s + 1.16i·7-s − 0.477·9-s + 0.300·11-s + 0.681·13-s + 0.602i·17-s + (−0.977 − 0.212i)19-s − 0.845i·21-s − 0.0389i·23-s + 1.06·27-s + 0.617i·29-s + 1.31i·31-s − 0.217·33-s − 0.207·37-s − 0.492·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.968 + 0.247i$
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: \((2,\ 1900,\ (\ :1),\ -0.968 + 0.247i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3699355340\)
\(L(\frac12)\) \(\approx\) \(0.3699355340\)
\(L(2)\) 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 \( 1 + (18.5 + 4.03i)T \)
good3 \( 1 + 2.16T + 9T^{2} \)
7 \( 1 - 8.18iT - 49T^{2} \)
11 \( 1 - 3.30T + 121T^{2} \)
13 \( 1 - 8.86T + 169T^{2} \)
17 \( 1 - 10.2iT - 289T^{2} \)
23 \( 1 + 0.896iT - 529T^{2} \)
29 \( 1 - 17.9iT - 841T^{2} \)
31 \( 1 - 40.8iT - 961T^{2} \)
37 \( 1 + 7.68T + 1.36e3T^{2} \)
41 \( 1 - 69.9iT - 1.68e3T^{2} \)
43 \( 1 + 25.9iT - 1.84e3T^{2} \)
47 \( 1 + 12.4iT - 2.20e3T^{2} \)
53 \( 1 - 55.9T + 2.80e3T^{2} \)
59 \( 1 + 105. iT - 3.48e3T^{2} \)
61 \( 1 + 48.2T + 3.72e3T^{2} \)
67 \( 1 + 14.6T + 4.48e3T^{2} \)
71 \( 1 - 2.89iT - 5.04e3T^{2} \)
73 \( 1 + 75.9iT - 5.32e3T^{2} \)
79 \( 1 - 86.1iT - 6.24e3T^{2} \)
83 \( 1 + 52.0iT - 6.88e3T^{2} \)
89 \( 1 - 39.5iT - 7.92e3T^{2} \)
97 \( 1 - 172.T + 9.40e3T^{2} \)
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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

−9.263945695502873946517024701278, −8.661288324023572081217905964105, −8.154568516497927387934983570594, −6.73503327843255533006488624079, −6.26779002395113186877377768395, −5.50831170954017018892851127137, −4.80551818635053944118774001514, −3.60612224255548341910815932279, −2.59701682176847454845358056040, −1.43203749973932901059864287894, 0.12047912879820135532771622162, 1.07824769865890791277794244692, 2.51491649931538184542073315299, 3.81873499358465531301290892890, 4.37847463388070490439020181038, 5.51912799863523025135132946388, 6.18040525271381402530012557656, 6.94928073291881460217338384905, 7.73663909158252872455537608923, 8.615709051675246450730743356282

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