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Properties

Label 2-1900-380.227-c0-0-6
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  + (0.923 − 0.382i)2-s + (−1.30 + 1.30i)3-s + (0.707 − 0.707i)4-s + (−0.707 + 1.70i)6-s + (0.382 − 0.923i)8-s − 2.41i·9-s − 1.41i·11-s + 1.84i·12-s + (−1.30 − 1.30i)13-s − i·16-s + (−0.923 − 2.23i)18-s − 19-s + (−0.541 − 1.30i)22-s + (0.707 + 1.70i)24-s + (−1.70 − 0.707i)26-s + (1.84 + 1.84i)27-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (−1.30 + 1.30i)3-s + (0.707 − 0.707i)4-s + (−0.707 + 1.70i)6-s + (0.382 − 0.923i)8-s − 2.41i·9-s − 1.41i·11-s + 1.84i·12-s + (−1.30 − 1.30i)13-s − i·16-s + (−0.923 − 2.23i)18-s − 19-s + (−0.541 − 1.30i)22-s + (0.707 + 1.70i)24-s + (−1.70 − 0.707i)26-s + (1.84 + 1.84i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.010232332\)
\(L(\frac12)\) \(\approx\) \(1.010232332\)
\(L(1)\) 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 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
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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

−9.722267322658238522240236291688, −8.675227021650885560463504803050, −7.42205601884610892574209799138, −6.29208410174247330503195967262, −5.78943032770958734435108918758, −5.16557802297764306155592322587, −4.42998549005147472519547947383, −3.58487533245122570704102365409, −2.71278323377652451438799651603, −0.61325165660979778554563150180, 1.85218336235448255564326332719, 2.35463293718713129196067622440, 4.28196597036317042190303012236, 4.81708959969662501035446230896, 5.59987524593255436979246764292, 6.62812838172203474209261624383, 6.89937560559978576226876197866, 7.48669257367086405767333819839, 8.375152746993087340521944653875, 9.725284115061841409299680301369

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