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Properties

Label 2-1900-95.18-c1-0-8
Degree $2$
Conductor $1900$
Sign $-0.597 - 0.801i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.49 + 1.49i)3-s + (−0.311 + 0.311i)7-s − 1.47i·9-s + 2.90·11-s + (−2.84 + 2.84i)13-s + (2.52 − 2.52i)17-s + (4.34 − 0.377i)19-s − 0.930i·21-s + (4.11 + 4.11i)23-s + (−2.28 − 2.28i)27-s − 2.99·29-s − 0.930i·31-s + (−4.34 + 4.34i)33-s + (8.11 + 8.11i)37-s − 8.51i·39-s + ⋯
L(s)  = 1  + (−0.863 + 0.863i)3-s + (−0.117 + 0.117i)7-s − 0.491i·9-s + 0.875·11-s + (−0.789 + 0.789i)13-s + (0.612 − 0.612i)17-s + (0.996 − 0.0866i)19-s − 0.203i·21-s + (0.858 + 0.858i)23-s + (−0.439 − 0.439i)27-s − 0.555·29-s − 0.167i·31-s + (−0.755 + 0.755i)33-s + (1.33 + 1.33i)37-s − 1.36i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.597 - 0.801i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.597 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086915942\)
\(L(\frac12)\) \(\approx\) \(1.086915942\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.34 + 0.377i)T \)
good3 \( 1 + (1.49 - 1.49i)T - 3iT^{2} \)
7 \( 1 + (0.311 - 0.311i)T - 7iT^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 + (2.84 - 2.84i)T - 13iT^{2} \)
17 \( 1 + (-2.52 + 2.52i)T - 17iT^{2} \)
23 \( 1 + (-4.11 - 4.11i)T + 23iT^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 + 0.930iT - 31T^{2} \)
37 \( 1 + (-8.11 - 8.11i)T + 37iT^{2} \)
41 \( 1 - 2.06iT - 41T^{2} \)
43 \( 1 + (2.11 + 2.11i)T + 43iT^{2} \)
47 \( 1 + (-2.73 + 2.73i)T - 47iT^{2} \)
53 \( 1 + (0.565 - 0.565i)T - 53iT^{2} \)
59 \( 1 + 9.32T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + (0.144 + 0.144i)T + 67iT^{2} \)
71 \( 1 - 9.61iT - 71T^{2} \)
73 \( 1 + (-4.09 - 4.09i)T + 73iT^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (9.21 + 9.21i)T + 83iT^{2} \)
89 \( 1 - 7.55T + 89T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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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

−9.556006508180426232459906928385, −9.110710928666148101982153187206, −7.78601796180995454976084413077, −7.08425745064635501813218059245, −6.17295720947575986135207933164, −5.32967851505449639933279624427, −4.72836837040179480164104797771, −3.86206601618005024220891365013, −2.79192991633008723189216285239, −1.23543733161376116338255634726, 0.51832180000096812932993913772, 1.51041281053989856775555401541, 2.89425719720678292080559211417, 3.95580523707629029893761984427, 5.13796961967034545845318808430, 5.81405495223526923088521228834, 6.55364919863104412325027747395, 7.31967756285722370238996939036, 7.85671727117165685377565834136, 9.006782188490739452575514907079

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