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Properties

Label 2-731-17.9-c1-0-33
Degree $2$
Conductor $731$
Sign $0.0758 + 0.997i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 1.70i)2-s + (−1 − 0.414i)3-s − 3.82i·4-s + (−0.585 + 1.41i)5-s + (2.41 − i)6-s + (0.414 + i)7-s + (3.12 + 3.12i)8-s + (−1.29 − 1.29i)9-s + (−1.41 − 3.41i)10-s + (1.70 − 0.707i)11-s + (−1.58 + 3.82i)12-s + 6.82i·13-s + (−2.41 − i)14-s + (1.17 − 1.17i)15-s − 2.99·16-s + (−0.121 − 4.12i)17-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (−0.577 − 0.239i)3-s − 1.91i·4-s + (−0.261 + 0.632i)5-s + (0.985 − 0.408i)6-s + (0.156 + 0.377i)7-s + (1.10 + 1.10i)8-s + (−0.430 − 0.430i)9-s + (−0.447 − 1.07i)10-s + (0.514 − 0.213i)11-s + (−0.457 + 1.10i)12-s + 1.89i·13-s + (−0.645 − 0.267i)14-s + (0.302 − 0.302i)15-s − 0.749·16-s + (−0.0294 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0758 + 0.997i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.0758 + 0.997i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.121 + 4.12i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.70 - 1.70i)T - 2iT^{2} \)
3 \( 1 + (1 + 0.414i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.585 - 1.41i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.414 - i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.70 + 0.707i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 6.82iT - 13T^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 + (4.12 - 1.70i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (5.12 + 2.12i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (6.24 + 2.58i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.87 + 4.53i)T + (-28.9 + 28.9i)T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + (-0.171 + 0.171i)T - 53iT^{2} \)
59 \( 1 + (7.48 + 7.48i)T + 59iT^{2} \)
61 \( 1 + (-0.585 - 1.41i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 7.07T + 67T^{2} \)
71 \( 1 + (8.07 + 3.34i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.75 - 9.07i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.53 + 3.94i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (6.82 - 6.82i)T - 83iT^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 + (-3.77 + 9.12i)T + (-68.5 - 68.5i)T^{2} \)
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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

−9.871802299033076897593221702282, −9.046963283458840344618344223595, −8.619359794489866955161841137453, −7.27991930792491930005995485140, −6.88073174401892637654144615652, −6.13502362086243651298704364654, −5.28100550548386681609480212992, −3.72999970128369109803767404523, −1.80476335188985763961081751060, 0, 1.26793441288875398764519362076, 2.71091861049034862228402428296, 3.90781461364585711173254420310, 5.01202075738896497441508316617, 6.14489389078576918637585420849, 7.68201996003824458037252210830, 8.273958729508280908483652661411, 8.909515887958966480254389887698, 10.06352039118500367626616728228

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