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Properties

Label 2-1900-95.18-c1-0-24
Degree $2$
Conductor $1900$
Sign $-0.977 + 0.212i$
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  + (0.880 − 0.880i)3-s + (−1.04 + 1.04i)7-s + 1.44i·9-s − 2.44·11-s + (−3.03 + 3.03i)13-s + (4.66 − 4.66i)17-s + (−4.11 − 1.44i)19-s + 1.84i·21-s + (−3.61 − 3.61i)23-s + (3.91 + 3.91i)27-s − 8.22·29-s − 10.0i·31-s + (−2.15 + 2.15i)33-s + (−5.19 − 5.19i)37-s + 5.34i·39-s + ⋯
L(s)  = 1  + (0.508 − 0.508i)3-s + (−0.396 + 0.396i)7-s + 0.483i·9-s − 0.738·11-s + (−0.842 + 0.842i)13-s + (1.13 − 1.13i)17-s + (−0.943 − 0.332i)19-s + 0.403i·21-s + (−0.754 − 0.754i)23-s + (0.753 + 0.753i)27-s − 1.52·29-s − 1.80i·31-s + (−0.375 + 0.375i)33-s + (−0.853 − 0.853i)37-s + 0.856i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.977 + 0.212i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3700469813\)
\(L(\frac12)\) \(\approx\) \(0.3700469813\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 + 1.44i)T \)
good3 \( 1 + (-0.880 + 0.880i)T - 3iT^{2} \)
7 \( 1 + (1.04 - 1.04i)T - 7iT^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + (3.03 - 3.03i)T - 13iT^{2} \)
17 \( 1 + (-4.66 + 4.66i)T - 17iT^{2} \)
23 \( 1 + (3.61 + 3.61i)T + 23iT^{2} \)
29 \( 1 + 8.22T + 29T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + (5.19 + 5.19i)T + 37iT^{2} \)
41 \( 1 + 1.84iT - 41T^{2} \)
43 \( 1 + (5.71 + 5.71i)T + 43iT^{2} \)
47 \( 1 + (-1.04 + 1.04i)T - 47iT^{2} \)
53 \( 1 + (-6.95 + 6.95i)T - 53iT^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 9.34T + 61T^{2} \)
67 \( 1 + (6.55 + 6.55i)T + 67iT^{2} \)
71 \( 1 - 1.84iT - 71T^{2} \)
73 \( 1 + (-9.33 - 9.33i)T + 73iT^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + (-12.4 - 12.4i)T + 83iT^{2} \)
89 \( 1 + 8.22T + 89T^{2} \)
97 \( 1 + (-10.4 - 10.4i)T + 97iT^{2} \)
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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

−8.902299887494742469272977736811, −7.84242709474321502888601692943, −7.50521530497974128051940483324, −6.61390499310744724341145249987, −5.58170511239808344287226808749, −4.86253724816075086038833248724, −3.72778038091622530209910452343, −2.47848369065338530362174494524, −2.10604419313420433155918270336, −0.11436679933777393495173407920, 1.65976919461724403039862028708, 3.13277291384905158562326663440, 3.51850407633919846387870033592, 4.60595761466250532707032494699, 5.57456535351584009319102825337, 6.33506960764077299906479848847, 7.43350936194711845806342054674, 8.022170620219970318563425612715, 8.803853569895327737561026626907, 9.698945172256413288742545181860

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