Properties

Label 2-736-184.101-c1-0-10
Degree $2$
Conductor $736$
Sign $0.747 - 0.664i$
Analytic cond. $5.87698$
Root an. cond. $2.42425$
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  + (1.56 − 1.35i)3-s + (−1.30 + 0.188i)5-s + (−1.93 + 4.22i)7-s + (0.180 − 1.25i)9-s + (−0.917 + 3.12i)11-s + (4.99 − 2.28i)13-s + (−1.78 + 2.06i)15-s + (3.05 + 1.96i)17-s + (1.11 + 1.73i)19-s + (2.70 + 9.21i)21-s + (4.20 − 2.30i)23-s + (−3.12 + 0.916i)25-s + (1.93 + 3.00i)27-s + (0.0805 − 0.125i)29-s + (−5.26 + 6.07i)31-s + ⋯
L(s)  = 1  + (0.901 − 0.781i)3-s + (−0.584 + 0.0840i)5-s + (−0.729 + 1.59i)7-s + (0.0602 − 0.419i)9-s + (−0.276 + 0.941i)11-s + (1.38 − 0.632i)13-s + (−0.461 + 0.532i)15-s + (0.741 + 0.476i)17-s + (0.255 + 0.397i)19-s + (0.590 + 2.01i)21-s + (0.876 − 0.481i)23-s + (−0.624 + 0.183i)25-s + (0.371 + 0.578i)27-s + (0.0149 − 0.0232i)29-s + (−0.945 + 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57385 + 0.598000i\)
\(L(\frac12)\) \(\approx\) \(1.57385 + 0.598000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-4.20 + 2.30i)T \)
good3 \( 1 + (-1.56 + 1.35i)T + (0.426 - 2.96i)T^{2} \)
5 \( 1 + (1.30 - 0.188i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (1.93 - 4.22i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (0.917 - 3.12i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-4.99 + 2.28i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-3.05 - 1.96i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-1.11 - 1.73i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-0.0805 + 0.125i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (5.26 - 6.07i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-6.57 - 0.945i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.661 + 4.60i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (4.20 - 3.64i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 2.17T + 47T^{2} \)
53 \( 1 + (-1.56 - 0.715i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-10.3 + 4.71i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (0.788 + 0.683i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (1.96 + 6.68i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (4.09 - 1.20i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (5.31 - 3.41i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (3.36 + 7.37i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-3.79 - 0.545i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (5.49 + 6.34i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-2.47 - 17.2i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.39801743164837727923700702000, −9.373629269959380014263076375259, −8.583085439302572552413123072105, −8.053892348483188657010160928811, −7.15459050377777670398276348522, −6.13076700410981316486824329393, −5.24428529491842396137691510510, −3.56871624002559602839315095604, −2.85393480752252712791274480419, −1.68754199042455785270840806708, 0.834349382225430134681237676099, 3.11901909732812509174848144457, 3.70041797699962206052129840103, 4.31850658349879159993030492208, 5.84790639457336028435715584711, 6.96440947994982553236788869177, 7.79803775161991496831007287860, 8.638037138989129340736634115345, 9.467942272387145281714138180601, 10.12463611907394664918512745185

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