Properties

Label 2-731-731.67-c1-0-9
Degree $2$
Conductor $731$
Sign $-0.0938 - 0.995i$
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  + (−0.792 + 0.993i)2-s + (−0.413 − 2.74i)3-s + (0.0856 + 0.375i)4-s + (1.48 + 2.17i)5-s + (3.05 + 1.76i)6-s + (0.708 − 0.408i)7-s + (−2.73 − 1.31i)8-s + (−4.48 + 1.38i)9-s + (−3.34 − 0.250i)10-s + (2.11 + 0.481i)11-s + (0.993 − 0.389i)12-s + (0.498 + 6.64i)13-s + (−0.154 + 1.02i)14-s + (5.36 − 4.97i)15-s + (2.77 − 1.33i)16-s + (−2.84 + 2.98i)17-s + ⋯
L(s)  = 1  + (−0.560 + 0.702i)2-s + (−0.238 − 1.58i)3-s + (0.0428 + 0.187i)4-s + (0.664 + 0.974i)5-s + (1.24 + 0.719i)6-s + (0.267 − 0.154i)7-s + (−0.965 − 0.464i)8-s + (−1.49 + 0.461i)9-s + (−1.05 − 0.0792i)10-s + (0.636 + 0.145i)11-s + (0.286 − 0.112i)12-s + (0.138 + 1.84i)13-s + (−0.0413 + 0.274i)14-s + (1.38 − 1.28i)15-s + (0.694 − 0.334i)16-s + (−0.690 + 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.0938 - 0.995i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.0938 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647085 + 0.710941i\)
\(L(\frac12)\) \(\approx\) \(0.647085 + 0.710941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.84 - 2.98i)T \)
43 \( 1 + (-2.61 + 6.01i)T \)
good2 \( 1 + (0.792 - 0.993i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.413 + 2.74i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-1.48 - 2.17i)T + (-1.82 + 4.65i)T^{2} \)
7 \( 1 + (-0.708 + 0.408i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.11 - 0.481i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.498 - 6.64i)T + (-12.8 + 1.93i)T^{2} \)
19 \( 1 + (4.49 + 1.38i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-0.451 + 0.486i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.901 - 5.98i)T + (-27.7 - 8.54i)T^{2} \)
31 \( 1 + (2.14 - 0.841i)T + (22.7 - 21.0i)T^{2} \)
37 \( 1 + (-4.64 - 2.68i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.17 - 1.73i)T + (9.12 + 39.9i)T^{2} \)
47 \( 1 + (-1.69 - 7.43i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.262 - 3.50i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (6.14 - 2.95i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.443 + 0.173i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (-8.36 - 2.58i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-4.28 - 4.62i)T + (-5.30 + 70.8i)T^{2} \)
73 \( 1 + (-11.0 + 0.831i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (-4.76 + 2.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.62 - 0.847i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (4.58 - 0.691i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-12.4 - 2.84i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.82279611819387637336008585120, −9.369453824842315648949496761556, −8.749467116205165541941039280996, −7.78176078800592317909219219492, −6.77376684555008101439145899170, −6.71778219204659094999277982420, −6.04184668088802126469697409040, −4.16185075704456774121856145141, −2.55209225310435681408126059340, −1.61942218887384188372374509680, 0.61169835535718816323620962589, 2.25578767223985981457988621735, 3.58842281460986736905483101308, 4.77773064369419314553777488769, 5.47533303952880404600820469132, 6.15082722249421267528243709257, 8.161224883428200694528025493474, 8.896176206113698279657455205520, 9.542449397051151923766279863166, 10.01921513923481464248106024643

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