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Properties

Label 2-1900-19.18-c2-0-19
Degree $2$
Conductor $1900$
Sign $-0.886 - 0.462i$
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  + 3.53i·3-s − 0.468·7-s − 3.51·9-s − 14.1·11-s + 11.0i·13-s + 22.0·17-s + (16.8 + 8.79i)19-s − 1.65i·21-s + 41.1·23-s + 19.4i·27-s + 46.7i·29-s − 33.9i·31-s − 49.9i·33-s − 62.6i·37-s − 39.2·39-s + ⋯
L(s)  = 1  + 1.17i·3-s − 0.0669·7-s − 0.390·9-s − 1.28·11-s + 0.853i·13-s + 1.29·17-s + (0.886 + 0.462i)19-s − 0.0789i·21-s + 1.79·23-s + 0.718i·27-s + 1.61i·29-s − 1.09i·31-s − 1.51i·33-s − 1.69i·37-s − 1.00·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.886 - 0.462i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ -0.886 - 0.462i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.680942126\)
\(L(\frac12)\) \(\approx\) \(1.680942126\)
\(L(2)\) 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 \( 1 + (-16.8 - 8.79i)T \)
good3 \( 1 - 3.53iT - 9T^{2} \)
7 \( 1 + 0.468T + 49T^{2} \)
11 \( 1 + 14.1T + 121T^{2} \)
13 \( 1 - 11.0iT - 169T^{2} \)
17 \( 1 - 22.0T + 289T^{2} \)
23 \( 1 - 41.1T + 529T^{2} \)
29 \( 1 - 46.7iT - 841T^{2} \)
31 \( 1 + 33.9iT - 961T^{2} \)
37 \( 1 + 62.6iT - 1.36e3T^{2} \)
41 \( 1 - 23.5iT - 1.68e3T^{2} \)
43 \( 1 - 32.8T + 1.84e3T^{2} \)
47 \( 1 + 43.2T + 2.20e3T^{2} \)
53 \( 1 - 99.9iT - 2.80e3T^{2} \)
59 \( 1 - 66.7iT - 3.48e3T^{2} \)
61 \( 1 + 36.0T + 3.72e3T^{2} \)
67 \( 1 - 93.5iT - 4.48e3T^{2} \)
71 \( 1 + 124. iT - 5.04e3T^{2} \)
73 \( 1 + 41.3T + 5.32e3T^{2} \)
79 \( 1 + 27.2iT - 6.24e3T^{2} \)
83 \( 1 - 32.9T + 6.88e3T^{2} \)
89 \( 1 + 86.3iT - 7.92e3T^{2} \)
97 \( 1 - 20.8iT - 9.40e3T^{2} \)
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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

−9.369494811031487631578943588519, −8.906583541184631713365828624312, −7.66419733919097607663704271120, −7.25862115773937259782503935010, −5.88161091062296072847661349286, −5.20272912180969723411025595319, −4.55637078964239157700563159751, −3.50455534499964395004627154481, −2.83331366334009545110039785513, −1.27493444341473184512603634076, 0.47728884697713650624584735223, 1.37566344418881630913858775763, 2.69878362390358437987339822333, 3.27069065867964965617689574289, 4.94440903971844045715200829009, 5.42450753979492619827121185993, 6.49404488132529984578266796764, 7.19512232784276972788572921011, 7.954674915841026474435166694227, 8.250662857371704571119566616397

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