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Properties

Label 2-1900-380.227-c0-0-3
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: $\chi_{1900} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6494161472\)
\(L(\frac12)\) \(\approx\) \(0.6494161472\)
\(L(1)\) 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

−9.388399802422476554199439445139, −8.849196008450423118874141231747, −7.978535824984664317747162762751, −7.31968235151780471159515519301, −6.14786424075969344051666386480, −5.64842048395329302014510547967, −4.36462260489670454478605398195, −3.48903225471807521387987812545, −2.45836522890461094581123297629, −1.01674185287284018880123620885, 0.949093783544591426102224395338, 2.01821344371936332339271269048, 3.33533752556147130701904857578, 4.77723360698260479241421604341, 5.63599329613040412937051813787, 6.42522222598444142388207440918, 7.06470823500005681519446598452, 7.70751179409688044255975435739, 8.503530672775383438633910433876, 9.533740335761332230391785440809

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