Properties

Label 2-731-731.67-c1-0-18
Degree $2$
Conductor $731$
Sign $-0.502 - 0.864i$
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  + (−1.11 + 1.39i)2-s + (0.311 + 2.06i)3-s + (−0.264 − 1.15i)4-s + (−1.52 − 2.23i)5-s + (−3.22 − 1.86i)6-s + (2.35 − 1.36i)7-s + (−1.30 − 0.628i)8-s + (−1.29 + 0.399i)9-s + (4.80 + 0.360i)10-s + (4.82 + 1.10i)11-s + (2.30 − 0.905i)12-s + (−0.0244 − 0.326i)13-s + (−0.724 + 4.80i)14-s + (4.12 − 3.83i)15-s + (4.47 − 2.15i)16-s + (−1.71 + 3.75i)17-s + ⋯
L(s)  = 1  + (−0.787 + 0.987i)2-s + (0.179 + 1.19i)3-s + (−0.132 − 0.579i)4-s + (−0.679 − 0.997i)5-s + (−1.31 − 0.760i)6-s + (0.891 − 0.514i)7-s + (−0.461 − 0.222i)8-s + (−0.431 + 0.133i)9-s + (1.51 + 0.113i)10-s + (1.45 + 0.332i)11-s + (0.666 − 0.261i)12-s + (−0.00678 − 0.0904i)13-s + (−0.193 + 1.28i)14-s + (1.06 − 0.989i)15-s + (1.11 − 0.538i)16-s + (−0.414 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545492 + 0.947606i\)
\(L(\frac12)\) \(\approx\) \(0.545492 + 0.947606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.71 - 3.75i)T \)
43 \( 1 + (-2.01 + 6.24i)T \)
good2 \( 1 + (1.11 - 1.39i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.311 - 2.06i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (1.52 + 2.23i)T + (-1.82 + 4.65i)T^{2} \)
7 \( 1 + (-2.35 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.82 - 1.10i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.0244 + 0.326i)T + (-12.8 + 1.93i)T^{2} \)
19 \( 1 + (0.176 + 0.0545i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-0.368 + 0.397i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.42 - 9.42i)T + (-27.7 - 8.54i)T^{2} \)
31 \( 1 + (-5.08 + 1.99i)T + (22.7 - 21.0i)T^{2} \)
37 \( 1 + (-8.80 - 5.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.50 + 5.98i)T + (9.12 + 39.9i)T^{2} \)
47 \( 1 + (-2.03 - 8.90i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.859 + 11.4i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-11.5 + 5.58i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-9.08 - 3.56i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (13.8 + 4.27i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-7.84 - 8.45i)T + (-5.30 + 70.8i)T^{2} \)
73 \( 1 + (10.4 - 0.779i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (-0.314 + 0.181i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.35 - 0.656i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-6.31 + 0.951i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-10.5 - 2.40i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.34728854105531107487705770795, −9.540041161794549655193862694128, −8.673094842189323052154471915293, −8.469260563830044963008653109510, −7.37921675024933249172990465317, −6.50272125926001014612454646614, −5.13648311055997480408648141211, −4.30842290358758994665970024941, −3.68649642196076746060153955123, −1.18037045452013710004393962013, 0.934121325061126062134677535495, 2.07727800239390843913302088914, 2.91271903108891940588981632887, 4.25097419374213419971351417168, 5.98700592850062510685861318315, 6.79919050913288163733576889007, 7.68394904096131115083386736738, 8.408022539619379421673637934008, 9.223306287118896885498890045383, 10.18162352775452059291357861232

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