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Properties

Label 2-736-184.13-c1-0-17
Degree $2$
Conductor $736$
Sign $0.681 + 0.731i$
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  + (1.35 − 2.11i)3-s + (3.81 + 1.74i)5-s + (−1.23 − 0.361i)7-s + (−1.37 − 3.00i)9-s + (2.10 + 1.82i)11-s + (−0.900 − 3.06i)13-s + (8.86 − 5.69i)15-s + (−0.200 − 1.39i)17-s + (−5.93 − 0.853i)19-s + (−2.43 + 2.11i)21-s + (4.77 − 0.469i)23-s + (8.26 + 9.53i)25-s + (−0.764 − 0.109i)27-s + (5.36 − 0.771i)29-s + (−2.59 + 1.66i)31-s + ⋯
L(s)  = 1  + (0.783 − 1.21i)3-s + (1.70 + 0.779i)5-s + (−0.465 − 0.136i)7-s + (−0.457 − 1.00i)9-s + (0.634 + 0.549i)11-s + (−0.249 − 0.850i)13-s + (2.29 − 1.47i)15-s + (−0.0486 − 0.338i)17-s + (−1.36 − 0.195i)19-s + (−0.531 + 0.460i)21-s + (0.995 − 0.0978i)23-s + (1.65 + 1.90i)25-s + (−0.147 − 0.0211i)27-s + (0.995 − 0.143i)29-s + (−0.465 + 0.299i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.681 + 0.731i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.681 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29526 - 0.998595i\)
\(L(\frac12)\) \(\approx\) \(2.29526 - 0.998595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.77 + 0.469i)T \)
good3 \( 1 + (-1.35 + 2.11i)T + (-1.24 - 2.72i)T^{2} \)
5 \( 1 + (-3.81 - 1.74i)T + (3.27 + 3.77i)T^{2} \)
7 \( 1 + (1.23 + 0.361i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-2.10 - 1.82i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (0.900 + 3.06i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.200 + 1.39i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (5.93 + 0.853i)T + (18.2 + 5.35i)T^{2} \)
29 \( 1 + (-5.36 + 0.771i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (2.59 - 1.66i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (6.97 - 3.18i)T + (24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.478 + 1.04i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (0.886 - 1.37i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 - 2.61T + 47T^{2} \)
53 \( 1 + (3.24 - 11.0i)T + (-44.5 - 28.6i)T^{2} \)
59 \( 1 + (2.45 + 8.35i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (4.47 + 6.95i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (6.96 - 6.03i)T + (9.53 - 66.3i)T^{2} \)
71 \( 1 + (-3.19 - 3.68i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.59 + 11.1i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (3.30 - 0.971i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (6.86 - 3.13i)T + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (-4.04 - 2.59i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (2.85 - 6.24i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.21029648958247634822172058733, −9.341830285553363954622329909333, −8.647812346919408941080956757334, −7.44471554260090300100500284728, −6.66115885439397686348499745880, −6.32296386804609988358000096558, −5.01032100426979091776215828509, −3.15154613135780572969992875754, −2.43195208168598898256057481306, −1.46754379987880506753618907345, 1.72597609059588517821518449784, 2.88050692145918180699751223732, 4.11593113796346456038957997721, 4.93839392723977909753846297335, 5.97365370873320272783435883533, 6.73188680300736796504942577860, 8.689128883431813772726151066100, 8.798141812600068724345695945233, 9.565490864623702439235544456367, 10.17398894751025915059686621189

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