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Properties

Label 2-731-43.35-c1-0-17
Degree $2$
Conductor $731$
Sign $-0.628 - 0.777i$
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  + (0.173 − 0.760i)2-s + (0.719 + 3.15i)3-s + (1.25 + 0.603i)4-s + (−1.24 + 1.56i)5-s + 2.52·6-s − 1.13·7-s + (1.64 − 2.06i)8-s + (−6.71 + 3.23i)9-s + (0.971 + 1.21i)10-s + (2.11 − 1.01i)11-s + (−1.00 + 4.38i)12-s + (−2.05 + 2.58i)13-s + (−0.196 + 0.860i)14-s + (−5.81 − 2.80i)15-s + (0.448 + 0.562i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.537i)2-s + (0.415 + 1.81i)3-s + (0.626 + 0.301i)4-s + (−0.556 + 0.698i)5-s + 1.02·6-s − 0.427·7-s + (0.583 − 0.731i)8-s + (−2.23 + 1.07i)9-s + (0.307 + 0.385i)10-s + (0.636 − 0.306i)11-s + (−0.289 + 1.26i)12-s + (−0.570 + 0.715i)13-s + (−0.0524 + 0.229i)14-s + (−1.50 − 0.723i)15-s + (0.112 + 0.140i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.628 - 0.777i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757022 + 1.58591i\)
\(L(\frac12)\) \(\approx\) \(0.757022 + 1.58591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (4.29 + 4.95i)T \)
good2 \( 1 + (-0.173 + 0.760i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.719 - 3.15i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.24 - 1.56i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 1.13T + 7T^{2} \)
11 \( 1 + (-2.11 + 1.01i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.05 - 2.58i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (1.39 + 0.671i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-2.60 + 1.25i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (1.52 - 6.69i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.50 + 6.57i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 + (0.435 - 1.90i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-0.368 - 0.177i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.42 + 3.04i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-7.38 - 9.26i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.25 + 5.49i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (2.23 + 1.07i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-8.94 - 4.30i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.514 + 0.644i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 6.20T + 79T^{2} \)
83 \( 1 + (-1.98 - 8.71i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.22 - 14.1i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-9.74 + 4.69i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.81589835454963118504313171619, −9.921940780243601307303491708376, −9.315509839377349891240638662797, −8.337728894280616199331358229471, −7.22676610743533465118688008847, −6.30046053723179257116776258797, −4.87210025163083365620069540996, −3.86650944020655441977985426433, −3.39262697057131293396482372960, −2.44306632065353041368554678862, 0.815951530225930111328897096537, 2.02766388391970119237663471360, 3.15161710420736980978182059109, 4.88026715499456167580659334130, 6.04886445609618394852486053039, 6.64131283846077358244440402581, 7.49602119311675790106511986124, 7.981803325127496191976820599119, 8.815165555008744857635536043849, 9.964616967969201828619519791422

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