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Properties

Label 2-1900-95.37-c1-0-26
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: $\chi_{1900} (1557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.977 - 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1855905515\)
\(L(\frac12)\) \(\approx\) \(0.1855905515\)
\(L(\frac{3}{2})\) 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

−8.975213927178550266379269348753, −8.007575730289859828577824985660, −6.97818533769442693848808949615, −6.55504379778578333947042654481, −5.62719164179976359621610200478, −4.82262918360547553776512551123, −3.84509158073424054446748944592, −2.58326398177102339459497341507, −1.56008981284264509733129550111, −0.07000386117310058156575972127, 1.64885260944374239301955283766, 2.84907047637361917720089181889, 4.16644237733478808993287610039, 4.59457200993837288701353970699, 5.76896079347214772399155653002, 6.11126398019816307280034443614, 7.53522121669688352908941947653, 7.936598700414408436887185953920, 8.846948770511315956067670400724, 9.726726414566688383158662750974

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