Properties

Label 2-1900-380.303-c0-0-5
Degree $2$
Conductor $1900$
Sign $0.973 + 0.229i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (0.541 + 0.541i)3-s + (0.707 − 0.707i)4-s + (0.707 + 0.292i)6-s + (0.382 − 0.923i)8-s − 0.414i·9-s + 1.41i·11-s + 0.765·12-s + (−0.541 + 0.541i)13-s i·16-s + (−0.158 − 0.382i)18-s + 19-s + (0.541 + 1.30i)22-s + (0.707 − 0.292i)24-s + (−0.292 + 0.707i)26-s + (0.765 − 0.765i)27-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.541 + 0.541i)3-s + (0.707 − 0.707i)4-s + (0.707 + 0.292i)6-s + (0.382 − 0.923i)8-s − 0.414i·9-s + 1.41i·11-s + 0.765·12-s + (−0.541 + 0.541i)13-s i·16-s + (−0.158 − 0.382i)18-s + 19-s + (0.541 + 1.30i)22-s + (0.707 − 0.292i)24-s + (−0.292 + 0.707i)26-s + (0.765 − 0.765i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.365630768\)
\(L(\frac12)\) \(\approx\) \(2.365630768\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
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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

−9.549056488400845637956126800767, −8.915385147966603273723071565371, −7.50172271964546948309979551305, −7.05121375546338107161080698818, −6.05368929955030553896256930501, −5.02130142531765024694614355813, −4.39820891415092260259942552080, −3.59628911183967796773882820332, −2.68297613835421237226703120875, −1.65932673570306424922656240100, 1.64253306514119950758168321757, 2.97423700743361914369795117728, 3.27265439433177827257583767795, 4.70277394112873954030995965499, 5.39080322661071122592310584816, 6.21569676264673889515586394503, 7.08528574953988993531879531992, 7.86551993188857415018544727521, 8.262211297894012234865409372782, 9.177496640036474792161557210132

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