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Properties

Label 2-731-731.16-c1-0-20
Degree $2$
Conductor $731$
Sign $-0.953 + 0.300i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  + (−0.594 − 2.60i)2-s + (−2.90 − 0.662i)3-s + (−4.62 + 2.22i)4-s + (−1.09 + 0.871i)5-s + 7.94i·6-s − 2.92i·7-s + (5.21 + 6.54i)8-s + (5.27 + 2.53i)9-s + (2.91 + 2.32i)10-s + (2.10 − 4.36i)11-s + (14.8 − 3.39i)12-s + (3.19 + 4.00i)13-s + (−7.62 + 1.74i)14-s + (3.74 − 1.80i)15-s + (7.53 − 9.44i)16-s + (4.12 + 0.0729i)17-s + ⋯
L(s)  = 1  + (−0.420 − 1.84i)2-s + (−1.67 − 0.382i)3-s + (−2.31 + 1.11i)4-s + (−0.488 + 0.389i)5-s + 3.24i·6-s − 1.10i·7-s + (1.84 + 2.31i)8-s + (1.75 + 0.846i)9-s + (0.922 + 0.735i)10-s + (0.633 − 1.31i)11-s + (4.29 − 0.981i)12-s + (0.884 + 1.10i)13-s + (−2.03 + 0.465i)14-s + (0.967 − 0.465i)15-s + (1.88 − 2.36i)16-s + (0.999 + 0.0176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0815011 - 0.529388i\)
\(L(\frac12)\) \(\approx\) \(0.0815011 - 0.529388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.12 - 0.0729i)T \)
43 \( 1 + (-2.45 + 6.08i)T \)
good2 \( 1 + (0.594 + 2.60i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (2.90 + 0.662i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.09 - 0.871i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.92iT - 7T^{2} \)
11 \( 1 + (-2.10 + 4.36i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.19 - 4.00i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-0.341 + 0.164i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.67 - 5.55i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (1.64 - 0.375i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-6.07 + 1.38i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 + (0.131 - 0.0300i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (7.04 - 3.39i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-6.70 + 8.40i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-9.15 + 11.4i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-9.83 - 2.24i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (7.30 - 3.51i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (1.50 + 3.12i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.644 + 0.514i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 7.39iT - 79T^{2} \)
83 \( 1 + (1.12 - 4.93i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.0415 + 0.182i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (3.93 - 8.17i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.37084318063599738183590316529, −9.688024968172462948850177180857, −8.472014515969984962601414884247, −7.46075930822577796988890370283, −6.43256341964317736412932465767, −5.29425017989598696314892890198, −3.96734813349464955327463607135, −3.58659951778288956829680759928, −1.48106346333287332161534498156, −0.70618257661645075853937195878, 0.827716610767117696648501429739, 4.18511303210839814629315094848, 4.86572433123078672582967184971, 5.76487312077550737261367424295, 6.12900520700824282521460925052, 7.10151285419612299965848674718, 8.084504283841007554857412365493, 8.831108676145108248424540774778, 9.916372298418378742355201802780, 10.33787091786364290545275504523

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