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Properties

Label 2-1900-95.18-c1-0-23
Degree $2$
Conductor $1900$
Sign $0.450 + 0.893i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.450 + 0.893i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.450 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.572004221\)
\(L(\frac12)\) \(\approx\) \(2.572004221\)
\(L(\frac{3}{2})\) 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

−8.746209036778644424796546880286, −8.487338978106677421203920715393, −7.42164927042062617972581947096, −6.96739200874379395088438300537, −6.00230849949442015439703153984, −5.11318596805143093234322736964, −3.79561824983758471548532374357, −3.08630217143449650716884278753, −2.02375350641487194391836885208, −0.982602516569116156717912729928, 1.33473805628746525279361945343, 2.64933427892789962311101790549, 3.71659808016227367871742005913, 4.09713522253697982178311777384, 5.10070183113868846566173385536, 6.44722938795844899459345463779, 6.73979583214404149296840301888, 8.243760718169885280553254955424, 8.585626546519932629661281253427, 9.259705500476523282810906611503

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