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Properties

Label 2-731-17.2-c1-0-4
Degree $2$
Conductor $731$
Sign $0.861 - 0.508i$
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  + (−0.768 − 0.768i)2-s + (−1.04 + 0.431i)3-s − 0.819i·4-s + (−0.778 − 1.88i)5-s + (1.13 + 0.468i)6-s + (0.510 − 1.23i)7-s + (−2.16 + 2.16i)8-s + (−1.22 + 1.22i)9-s + (−0.846 + 2.04i)10-s + (0.949 + 0.393i)11-s + (0.353 + 0.852i)12-s + 6.82i·13-s + (−1.34 + 0.555i)14-s + (1.62 + 1.62i)15-s + 1.69·16-s + (−3.44 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.543 − 0.543i)2-s + (−0.601 + 0.248i)3-s − 0.409i·4-s + (−0.348 − 0.840i)5-s + (0.461 + 0.191i)6-s + (0.193 − 0.466i)7-s + (−0.765 + 0.765i)8-s + (−0.407 + 0.407i)9-s + (−0.267 + 0.646i)10-s + (0.286 + 0.118i)11-s + (0.101 + 0.246i)12-s + 1.89i·13-s + (−0.358 + 0.148i)14-s + (0.418 + 0.418i)15-s + 0.422·16-s + (−0.836 + 0.547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479251 + 0.130922i\)
\(L(\frac12)\) \(\approx\) \(0.479251 + 0.130922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.44 - 2.25i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.768 + 0.768i)T + 2iT^{2} \)
3 \( 1 + (1.04 - 0.431i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.778 + 1.88i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.510 + 1.23i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.949 - 0.393i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 6.82iT - 13T^{2} \)
19 \( 1 + (4.38 + 4.38i)T + 19iT^{2} \)
23 \( 1 + (-6.49 - 2.69i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.399 + 0.964i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.647 + 0.268i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-8.78 + 3.64i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.14 - 7.59i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 + (6.10 + 6.10i)T + 53iT^{2} \)
59 \( 1 + (3.98 - 3.98i)T - 59iT^{2} \)
61 \( 1 + (3.84 - 9.29i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 8.01T + 67T^{2} \)
71 \( 1 + (8.03 - 3.32i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.46 - 3.52i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-8.74 - 3.62i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.98 - 5.98i)T + 83iT^{2} \)
89 \( 1 - 0.713iT - 89T^{2} \)
97 \( 1 + (6.91 + 16.7i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.73138167327800948221467138935, −9.471755422395366727766172205216, −9.016940045445112391724468887989, −8.231656523642867020252272872996, −6.83746227310094847353302523711, −6.04240400869869175069358890665, −4.64680345999382665666224981618, −4.48764309370004542981980163761, −2.40980141569430372149765002559, −1.15493083261308505556546327859, 0.38097186674948005477982181797, 2.77788396560269284523199333847, 3.55855040730406532082821209419, 5.17073457474733520010896289616, 6.23054367265666910002612403747, 6.77052229585826218701581729237, 7.71880162447008063432711686888, 8.479126182098341900347950596060, 9.195856595376421411525075076189, 10.48762493406282342995395314146

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