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Properties

Label 2-1900-95.94-c2-0-44
Degree $2$
Conductor $1900$
Sign $0.366 + 0.930i$
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  + 1.84·3-s − 10.5i·7-s − 5.60·9-s + 12.7·11-s + 23.0·13-s + 4.77i·17-s + (−1.66 + 18.9i)19-s − 19.4i·21-s − 35.7i·23-s − 26.9·27-s + 13.7i·29-s − 30.5i·31-s + 23.4·33-s + 54.8·37-s + 42.4·39-s + ⋯
L(s)  = 1  + 0.614·3-s − 1.50i·7-s − 0.622·9-s + 1.15·11-s + 1.77·13-s + 0.280i·17-s + (−0.0878 + 0.996i)19-s − 0.924i·21-s − 1.55i·23-s − 0.996·27-s + 0.474i·29-s − 0.984i·31-s + 0.710·33-s + 1.48·37-s + 1.08·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.366 + 0.930i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ 0.366 + 0.930i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.732002366\)
\(L(\frac12)\) \(\approx\) \(2.732002366\)
\(L(2)\) 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 \( 1 + (1.66 - 18.9i)T \)
good3 \( 1 - 1.84T + 9T^{2} \)
7 \( 1 + 10.5iT - 49T^{2} \)
11 \( 1 - 12.7T + 121T^{2} \)
13 \( 1 - 23.0T + 169T^{2} \)
17 \( 1 - 4.77iT - 289T^{2} \)
23 \( 1 + 35.7iT - 529T^{2} \)
29 \( 1 - 13.7iT - 841T^{2} \)
31 \( 1 + 30.5iT - 961T^{2} \)
37 \( 1 - 54.8T + 1.36e3T^{2} \)
41 \( 1 - 18.6iT - 1.68e3T^{2} \)
43 \( 1 - 29.0iT - 1.84e3T^{2} \)
47 \( 1 + 11.3iT - 2.20e3T^{2} \)
53 \( 1 - 10.7T + 2.80e3T^{2} \)
59 \( 1 + 54.5iT - 3.48e3T^{2} \)
61 \( 1 - 17.7T + 3.72e3T^{2} \)
67 \( 1 + 60.2T + 4.48e3T^{2} \)
71 \( 1 - 39.9iT - 5.04e3T^{2} \)
73 \( 1 + 24.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.95iT - 6.24e3T^{2} \)
83 \( 1 + 130. iT - 6.88e3T^{2} \)
89 \( 1 + 160. iT - 7.92e3T^{2} \)
97 \( 1 + 71.2T + 9.40e3T^{2} \)
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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

−8.700681283681611321596360229832, −8.242942665734926893477842997178, −7.43519802036955439605177650777, −6.35400530610374092520890180327, −6.04268023159938613846336657144, −4.37962699497480404277746092322, −3.90121294257553545896762967679, −3.13702701191615718805707622473, −1.67164419270698421081884659646, −0.72042699276453932849047785031, 1.22412318560584485714408519628, 2.35698432441244501649204077689, 3.22217947177182845179530172247, 4.01226561877592244625092219609, 5.35791379436987296751536797574, 5.94935129840992564116726516936, 6.68201212404378761857188513187, 7.84952338744655848147933073098, 8.660699856178157668974558763073, 9.051644092732639881920641528104

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