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Properties

Label 2-731-43.35-c1-0-6
Degree $2$
Conductor $731$
Sign $-0.661 - 0.750i$
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  + (−0.277 + 1.21i)2-s + (−0.676 − 2.96i)3-s + (0.402 + 0.193i)4-s + (−0.276 + 0.346i)5-s + 3.78·6-s + 2.21·7-s + (−1.90 + 2.38i)8-s + (−5.62 + 2.70i)9-s + (−0.344 − 0.432i)10-s + (−4.65 + 2.24i)11-s + (0.302 − 1.32i)12-s + (−3.07 + 3.85i)13-s + (−0.614 + 2.69i)14-s + (1.21 + 0.585i)15-s + (−1.81 − 2.27i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.196 + 0.859i)2-s + (−0.390 − 1.71i)3-s + (0.201 + 0.0969i)4-s + (−0.123 + 0.155i)5-s + 1.54·6-s + 0.837·7-s + (−0.672 + 0.842i)8-s + (−1.87 + 0.902i)9-s + (−0.108 − 0.136i)10-s + (−1.40 + 0.675i)11-s + (0.0873 − 0.382i)12-s + (−0.851 + 1.06i)13-s + (−0.164 + 0.719i)14-s + (0.313 + 0.151i)15-s + (−0.452 − 0.567i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.661 - 0.750i$
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: \((2,\ 731,\ (\ :1/2),\ -0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.249493 + 0.552606i\)
\(L(\frac12)\) \(\approx\) \(0.249493 + 0.552606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.49 + 4.77i)T \)
good2 \( 1 + (0.277 - 1.21i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.676 + 2.96i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.276 - 0.346i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 - 2.21T + 7T^{2} \)
11 \( 1 + (4.65 - 2.24i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.07 - 3.85i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (1.86 + 0.899i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (3.99 - 1.92i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-1.22 + 5.36i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.130 - 0.572i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + (1.64 - 7.20i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (7.49 + 3.61i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-3.02 - 3.79i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (0.749 + 0.939i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-0.754 - 3.30i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-5.42 - 2.61i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (2.70 + 1.30i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (0.921 - 1.15i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 7.97T + 79T^{2} \)
83 \( 1 + (-1.17 - 5.15i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-1.71 - 7.51i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (17.0 - 8.23i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.09092369797211406341782990535, −9.722151347615411522063692519338, −8.275140172258278180477937526466, −7.894511359097380569420377602043, −7.26209275069555573715875385782, −6.60954238781497052747875296373, −5.68008996024668990599405091062, −4.78951211077401338219573604359, −2.55629777226782863611268652247, −1.86746798607033314019489869762, 0.31570529015917287816104405382, 2.50508831008190312119112177331, 3.35750287474483986887748747176, 4.59686713981855084149541731497, 5.26221181849509581024922981504, 6.15144227842491515272700017170, 7.85467992891642199846226751282, 8.554799746407114526742229432908, 9.753952187277781374950422825088, 10.23239450802622801380801333150

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