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Properties

Label 2-731-43.35-c1-0-19
Degree $2$
Conductor $731$
Sign $0.929 + 0.369i$
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  + (−0.370 + 1.62i)2-s + (−0.724 − 3.17i)3-s + (−0.692 − 0.333i)4-s + (−2.53 + 3.17i)5-s + 5.41·6-s − 2.21·7-s + (−1.27 + 1.60i)8-s + (−6.84 + 3.29i)9-s + (−4.20 − 5.27i)10-s + (2.88 − 1.39i)11-s + (−0.556 + 2.43i)12-s + (3.72 − 4.67i)13-s + (0.819 − 3.58i)14-s + (11.9 + 5.73i)15-s + (−3.08 − 3.86i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.261 + 1.14i)2-s + (−0.418 − 1.83i)3-s + (−0.346 − 0.166i)4-s + (−1.13 + 1.41i)5-s + 2.21·6-s − 0.836·7-s + (−0.451 + 0.566i)8-s + (−2.28 + 1.09i)9-s + (−1.33 − 1.66i)10-s + (0.870 − 0.419i)11-s + (−0.160 + 0.704i)12-s + (1.03 − 1.29i)13-s + (0.218 − 0.959i)14-s + (3.07 + 1.47i)15-s + (−0.771 − 0.966i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.929 + 0.369i$
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: \((2,\ 731,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693076 - 0.132870i\)
\(L(\frac12)\) \(\approx\) \(0.693076 - 0.132870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.16 - 2.22i)T \)
good2 \( 1 + (0.370 - 1.62i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.724 + 3.17i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.53 - 3.17i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.21T + 7T^{2} \)
11 \( 1 + (-2.88 + 1.39i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.72 + 4.67i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (-1.54 - 0.741i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.21 + 0.585i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.385 + 1.68i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.692 + 3.03i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 0.711T + 37T^{2} \)
41 \( 1 + (-2.06 + 9.02i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-8.97 - 4.32i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-6.84 - 8.58i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (5.95 + 7.46i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (2.83 + 12.4i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-0.820 - 0.395i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (6.55 + 3.15i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-7.39 + 9.27i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 1.38T + 79T^{2} \)
83 \( 1 + (-0.527 - 2.31i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.503 + 2.20i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (7.05 - 3.39i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.72300719652557788363556157511, −8.963468156783319387282058983946, −7.951818218204782211278154060154, −7.64089321362527395120105759136, −6.86709969942408869860851666517, −6.19102794097466199918985038842, −5.86873800285108791595711390818, −3.50287712871292903645329525934, −2.67783528978781022553394234125, −0.53979467656984799718641205901, 1.02442906284726203067240883103, 3.22010913486805341143112522320, 4.06298209394982824784539848594, 4.36014376954638352614616271568, 5.65059561236898868012862529863, 6.80900372957537121076681356735, 8.638140446422942015152016132855, 9.136487262496228903438993035555, 9.516344286535773636136836214744, 10.46640219126663275499434305241

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