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Properties

Label 2-2736-19.7-c1-0-25
Degree $2$
Conductor $2736$
Sign $0.950 - 0.310i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 + 2.02i)5-s − 0.538·7-s + 4.33·11-s + (−1.31 + 2.28i)13-s + (−1.53 − 2.66i)17-s + (−0.0218 − 4.35i)19-s + (4.40 − 7.62i)23-s + (−0.221 + 0.384i)25-s + (−0.699 + 1.21i)29-s + 5.57·31-s + (−0.628 − 1.08i)35-s + 12.1·37-s + (−0.838 − 1.45i)41-s + (4.32 + 7.48i)43-s + (2.79 − 4.84i)47-s + ⋯
L(s)  = 1  + (0.521 + 0.903i)5-s − 0.203·7-s + 1.30·11-s + (−0.365 + 0.632i)13-s + (−0.373 − 0.646i)17-s + (−0.00501 − 0.999i)19-s + (0.918 − 1.59i)23-s + (−0.0443 + 0.0768i)25-s + (−0.129 + 0.225i)29-s + 1.00·31-s + (−0.106 − 0.183i)35-s + 1.99·37-s + (−0.130 − 0.226i)41-s + (0.659 + 1.14i)43-s + (0.407 − 0.706i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.950 - 0.310i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.950 - 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.180227181\)
\(L(\frac12)\) \(\approx\) \(2.180227181\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (0.0218 + 4.35i)T \)
good5 \( 1 + (-1.16 - 2.02i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.538T + 7T^{2} \)
11 \( 1 - 4.33T + 11T^{2} \)
13 \( 1 + (1.31 - 2.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.53 + 2.66i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.40 + 7.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.699 - 1.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.57T + 31T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 + (0.838 + 1.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.32 - 7.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.79 + 4.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.96 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 + 1.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.34 + 7.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.36 - 4.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.17 - 5.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.16 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.41 + 4.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + (2.94 - 5.09i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.644 - 1.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.150888129226321131459381816642, −8.132915041078739964726407727014, −6.95364098512837922450390648347, −6.70218930292085298024220429962, −6.09850709982982077013997817323, −4.77966417012681797234786365817, −4.23070936410270981641819050326, −2.89595048547324468016837167370, −2.42901048165470622542624405952, −0.948641869257886382554499640902, 0.992194077932507746176267521042, 1.80985727893285791313005284229, 3.16606565982530462702745490572, 4.05463300720803817637214930613, 4.89892337068377951073579716298, 5.78750039614861025528244074858, 6.29926334735229006342950500616, 7.32978767163670688322998855260, 8.115108495596903807408057461452, 8.880716364479652024685236864305

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