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Properties

Label 2-731-43.6-c1-0-20
Degree $2$
Conductor $731$
Sign $0.945 + 0.326i$
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  − 0.884·2-s + (−1.14 + 1.98i)3-s − 1.21·4-s + (−1.47 + 2.56i)5-s + (1.01 − 1.75i)6-s + (−0.890 − 1.54i)7-s + 2.84·8-s + (−1.11 − 1.93i)9-s + (1.30 − 2.26i)10-s − 5.91·11-s + (1.39 − 2.41i)12-s + (−1.61 − 2.79i)13-s + (0.788 + 1.36i)14-s + (−3.38 − 5.86i)15-s − 0.0839·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 0.625·2-s + (−0.660 + 1.14i)3-s − 0.608·4-s + (−0.661 + 1.14i)5-s + (0.413 − 0.715i)6-s + (−0.336 − 0.583i)7-s + 1.00·8-s + (−0.372 − 0.645i)9-s + (0.413 − 0.716i)10-s − 1.78·11-s + (0.402 − 0.696i)12-s + (−0.447 − 0.775i)13-s + (0.210 + 0.364i)14-s + (−0.874 − 1.51i)15-s − 0.0209·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179424 - 0.0301629i\)
\(L(\frac12)\) \(\approx\) \(0.179424 - 0.0301629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-5.26 - 3.90i)T \)
good2 \( 1 + 0.884T + 2T^{2} \)
3 \( 1 + (1.14 - 1.98i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.47 - 2.56i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.890 + 1.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.91T + 11T^{2} \)
13 \( 1 + (1.61 + 2.79i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-1.92 + 3.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.29 - 3.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.37 - 5.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.41 - 4.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.72 + 6.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
47 \( 1 - 3.01T + 47T^{2} \)
53 \( 1 + (-2.64 + 4.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 + (-7.30 - 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.53 - 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.08 + 5.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.01 + 6.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.45 + 14.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.37 - 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.06 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−10.39480451123827081094001239224, −9.919855533656866604766008854302, −8.752659625446254833179485365328, −7.56242125361696383084530654716, −7.30515825559099391530316339512, −5.58884548310861186920138643065, −4.93079812164979406682549580112, −3.86335916925316723571705949959, −2.94652232195573064554008148070, −0.19406118931820259458797535677, 0.796886386103253531704473821377, 2.28829815715253469987773156023, 4.23476935202657463259260435980, 5.13700032944663911643522909532, 5.93636187316818700665246140037, 7.26728922329424154923527733320, 7.987487002409986561883583814895, 8.450379320949860770831118801429, 9.524721336915440699705414219614, 10.24917001944696546564005312418

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