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Properties

Label 2-1900-380.227-c0-0-0
Degree $2$
Conductor $1900$
Sign $-0.229 - 0.973i$
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  + (−0.382 − 0.923i)2-s + (−0.541 + 0.541i)3-s + (−0.707 + 0.707i)4-s + (0.707 + 0.292i)6-s + (0.923 + 0.382i)8-s + 0.414i·9-s + 1.41i·11-s − 0.765i·12-s + (−0.541 − 0.541i)13-s − i·16-s + (0.382 − 0.158i)18-s − 19-s + (1.30 − 0.541i)22-s + (−0.707 + 0.292i)24-s + (−0.292 + 0.707i)26-s + (−0.765 − 0.765i)27-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.541 + 0.541i)3-s + (−0.707 + 0.707i)4-s + (0.707 + 0.292i)6-s + (0.923 + 0.382i)8-s + 0.414i·9-s + 1.41i·11-s − 0.765i·12-s + (−0.541 − 0.541i)13-s − i·16-s + (0.382 − 0.158i)18-s − 19-s + (1.30 − 0.541i)22-s + (−0.707 + 0.292i)24-s + (−0.292 + 0.707i)26-s + (−0.765 − 0.765i)27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3877761227\)
\(L(\frac12)\) \(\approx\) \(0.3877761227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
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

−9.986678742571154276236645095291, −9.096519976602568315936798928236, −8.147101955261457568737478215988, −7.51392407277433178510816857005, −6.51382471474814745550431487502, −5.13748970487831157689641867481, −4.75696439551653255228287740965, −3.87814318723081949484059291431, −2.64282515902634475345637575612, −1.73855866296181391514161069727, 0.34300803069080927988521601884, 1.75290790538994615801277609397, 3.42856697223432367901445884125, 4.47674126739951430638698534366, 5.52964384902852039967394271784, 6.11506495281474142309961161283, 6.76263559246322000920699573525, 7.46077752104511353046476501859, 8.411297325572516574176361139304, 8.971705605492976818468406186685

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