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Properties

Label 2-2740-2740.19-c0-0-0
Degree $2$
Conductor $2740$
Sign $0.903 + 0.429i$
Analytic cond. $1.36743$
Root an. cond. $1.16937$
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  + (−0.932 + 0.361i)2-s + (0.739 − 0.673i)4-s + (0.273 − 0.961i)5-s + (−0.445 + 0.895i)8-s + (0.526 − 0.850i)9-s + (0.0922 + 0.995i)10-s + (1.97 − 0.276i)13-s + (0.0922 − 0.995i)16-s + (0.798 + 1.60i)17-s + (−0.183 + 0.982i)18-s + (−0.445 − 0.895i)20-s + (−0.850 − 0.526i)25-s + (−1.74 + 0.972i)26-s + (−0.212 + 0.176i)29-s + (0.273 + 0.961i)32-s + ⋯
L(s)  = 1  + (−0.932 + 0.361i)2-s + (0.739 − 0.673i)4-s + (0.273 − 0.961i)5-s + (−0.445 + 0.895i)8-s + (0.526 − 0.850i)9-s + (0.0922 + 0.995i)10-s + (1.97 − 0.276i)13-s + (0.0922 − 0.995i)16-s + (0.798 + 1.60i)17-s + (−0.183 + 0.982i)18-s + (−0.445 − 0.895i)20-s + (−0.850 − 0.526i)25-s + (−1.74 + 0.972i)26-s + (−0.212 + 0.176i)29-s + (0.273 + 0.961i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2740\)    =    \(2^{2} \cdot 5 \cdot 137\)
Sign: $0.903 + 0.429i$
Analytic conductor: \(1.36743\)
Root analytic conductor: \(1.16937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2740} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2740,\ (\ :0),\ 0.903 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9946484697\)
\(L(\frac12)\) \(\approx\) \(0.9946484697\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (-0.273 + 0.961i)T \)
137 \( 1 + (0.602 + 0.798i)T \)
good3 \( 1 + (-0.526 + 0.850i)T^{2} \)
7 \( 1 + (0.273 + 0.961i)T^{2} \)
11 \( 1 + (0.0922 - 0.995i)T^{2} \)
13 \( 1 + (-1.97 + 0.276i)T + (0.961 - 0.273i)T^{2} \)
17 \( 1 + (-0.798 - 1.60i)T + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (0.932 + 0.361i)T^{2} \)
23 \( 1 + (-0.183 + 0.982i)T^{2} \)
29 \( 1 + (0.212 - 0.176i)T + (0.183 - 0.982i)T^{2} \)
31 \( 1 + (0.361 + 0.932i)T^{2} \)
37 \( 1 + 0.722T + T^{2} \)
41 \( 1 + (-1.16 - 1.16i)T + iT^{2} \)
43 \( 1 + (0.361 - 0.932i)T^{2} \)
47 \( 1 + (-0.895 + 0.445i)T^{2} \)
53 \( 1 + (1.56 - 1.07i)T + (0.361 - 0.932i)T^{2} \)
59 \( 1 + (-0.445 - 0.895i)T^{2} \)
61 \( 1 + (-0.0971 - 0.156i)T + (-0.445 + 0.895i)T^{2} \)
67 \( 1 + (0.961 - 0.273i)T^{2} \)
71 \( 1 + (-0.995 + 0.0922i)T^{2} \)
73 \( 1 + (-0.840 + 0.634i)T + (0.273 - 0.961i)T^{2} \)
79 \( 1 + (-0.526 - 0.850i)T^{2} \)
83 \( 1 + (-0.798 + 0.602i)T^{2} \)
89 \( 1 + (-1.59 + 0.705i)T + (0.673 - 0.739i)T^{2} \)
97 \( 1 + (1.41 + 0.0653i)T + (0.995 + 0.0922i)T^{2} \)
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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

−8.931763837464868616807787929345, −8.254232255171439377443776858179, −7.77780811819739794398288417050, −6.47158549359036310198771042298, −6.13297120012238717185588560926, −5.39427974205822373623068610621, −4.15783727212836187743249520474, −3.33450755999888196095673498043, −1.62770394795543863969192143892, −1.12144480578584918643100594085, 1.29080583624903224233783474057, 2.29289892322562677420530301870, 3.21899351210034914931474176955, 3.97422821628512288319678228292, 5.34026337278277003269419363017, 6.27156475587830084275190574640, 6.95158913741203823334604497909, 7.64126445919910388769433673181, 8.251576022012102878183203616622, 9.267958335610982540068002045709

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