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Properties

Label 2-731-731.208-c1-0-50
Degree $2$
Conductor $731$
Sign $-0.0258 + 0.999i$
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  − 2.13i·2-s + (3.06 + 0.821i)3-s − 2.57·4-s + (−0.243 − 0.0653i)5-s + (1.75 − 6.56i)6-s + (3.12 − 0.837i)7-s + 1.23i·8-s + (6.13 + 3.54i)9-s + (−0.139 + 0.521i)10-s + (−0.891 − 0.891i)11-s + (−7.89 − 2.11i)12-s + (−1.42 + 2.46i)13-s + (−1.79 − 6.68i)14-s + (−0.694 − 0.401i)15-s − 2.51·16-s + (4.00 + 0.973i)17-s + ⋯
L(s)  = 1  − 1.51i·2-s + (1.77 + 0.474i)3-s − 1.28·4-s + (−0.109 − 0.0292i)5-s + (0.717 − 2.67i)6-s + (1.18 − 0.316i)7-s + 0.435i·8-s + (2.04 + 1.18i)9-s + (−0.0442 + 0.165i)10-s + (−0.268 − 0.268i)11-s + (−2.28 − 0.611i)12-s + (−0.395 + 0.684i)13-s + (−0.478 − 1.78i)14-s + (−0.179 − 0.103i)15-s − 0.629·16-s + (0.971 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0258 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.0258 + 0.999i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.0258 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98387 - 2.03592i\)
\(L(\frac12)\) \(\approx\) \(1.98387 - 2.03592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.00 - 0.973i)T \)
43 \( 1 + (-4.06 + 5.14i)T \)
good2 \( 1 + 2.13iT - 2T^{2} \)
3 \( 1 + (-3.06 - 0.821i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.243 + 0.0653i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.12 + 0.837i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.891 + 0.891i)T + 11iT^{2} \)
13 \( 1 + (1.42 - 2.46i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (4.11 - 2.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.514 + 1.92i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.11 - 4.14i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.77 - 0.742i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.12 + 2.17i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.58 + 6.58i)T + 41iT^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 + (-7.58 + 4.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.362iT - 59T^{2} \)
61 \( 1 + (9.95 - 2.66i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.04 + 7.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.83 - 10.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.03 - 3.84i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.5 + 2.83i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.12 - 0.651i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.69 - 2.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.62 - 7.62i)T - 97iT^{2} \)
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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

−10.45031709261029725724577670650, −9.433177212532674622994430377659, −8.589392275391768344302071437957, −8.109762765784309365258775961036, −7.06628843953718016258975200736, −4.99271347554423952098900508846, −4.10781684407211139468919469196, −3.49082979303219115506882595213, −2.30931757661735075877018991532, −1.62877056844879053670489129132, 1.82831864968655710919556433255, 2.98272374125853664114797639561, 4.39040400567656235774676306099, 5.28148763606231277328272403697, 6.52525802499824153849847039734, 7.57589688315722294114819891908, 7.85614581477594339955307980541, 8.444080889141592738906109696551, 9.245038870238571055859071549131, 10.15736371084025680717515772122

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