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Properties

Label 2-1900-95.94-c2-0-18
Degree $2$
Conductor $1900$
Sign $-0.447 - 0.894i$
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  + 8.82i·7-s − 9·9-s + 17.3·11-s + 33.9i·17-s + 19·19-s − 30i·23-s + 31.1i·43-s − 11.5i·47-s − 28.8·49-s − 108.·61-s − 79.4i·63-s + 137. i·73-s + 153. i·77-s + 81·81-s + 90i·83-s + ⋯
L(s)  = 1  + 1.26i·7-s − 9-s + 1.57·11-s + 1.99i·17-s + 19-s − 1.30i·23-s + 0.725i·43-s − 0.246i·47-s − 0.589·49-s − 1.77·61-s − 1.26i·63-s + 1.87i·73-s + 1.99i·77-s + 81-s + 1.08i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 1900,\ (\ :1),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.637992078\)
\(L(\frac12)\) \(\approx\) \(1.637992078\)
\(L(2)\) 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 \( 1 - 19T \)
good3 \( 1 + 9T^{2} \)
7 \( 1 - 8.82iT - 49T^{2} \)
11 \( 1 - 17.3T + 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 33.9iT - 289T^{2} \)
23 \( 1 + 30iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 31.1iT - 1.84e3T^{2} \)
47 \( 1 + 11.5iT - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + 108.T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 137. iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 90iT - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 9.40e3T^{2} \)
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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

−9.052333284661403527719205105885, −8.646589718687189536878003986663, −7.977540450017471918655036957307, −6.63946738744601313995562183086, −6.10487747646180884592202170228, −5.47112434530816358693177120893, −4.30594547665405074871127309417, −3.38153463634724409254475836073, −2.39830462726372269722621045547, −1.30373535583848900119308089167, 0.45288250904987640902847881987, 1.42747722735286990211846970394, 2.99380175250934575305332948287, 3.67019378559304485186281163144, 4.66315781078995263602336963644, 5.53099272534833963374954387871, 6.51569113508057661306949954467, 7.27926352759772441867305707083, 7.76351187778918195207792426526, 9.168255915948702844641858091679

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