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Properties

Label 2-731-731.16-c1-0-6
Degree $2$
Conductor $731$
Sign $0.590 - 0.807i$
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  + (−0.534 − 2.34i)2-s + (2.00 + 0.457i)3-s + (−3.39 + 1.63i)4-s + (−1.96 + 1.56i)5-s − 4.94i·6-s + 1.32i·7-s + (2.65 + 3.32i)8-s + (1.11 + 0.535i)9-s + (4.71 + 3.75i)10-s + (−0.0593 + 0.123i)11-s + (−7.56 + 1.72i)12-s + (−1.70 − 2.14i)13-s + (3.10 − 0.709i)14-s + (−4.65 + 2.23i)15-s + (1.66 − 2.08i)16-s + (−2.08 + 3.55i)17-s + ⋯
L(s)  = 1  + (−0.377 − 1.65i)2-s + (1.15 + 0.264i)3-s + (−1.69 + 0.817i)4-s + (−0.877 + 0.699i)5-s − 2.01i·6-s + 0.501i·7-s + (0.936 + 1.17i)8-s + (0.370 + 0.178i)9-s + (1.48 + 1.18i)10-s + (−0.0178 + 0.0371i)11-s + (−2.18 + 0.498i)12-s + (−0.473 − 0.593i)13-s + (0.830 − 0.189i)14-s + (−1.20 + 0.578i)15-s + (0.416 − 0.522i)16-s + (−0.504 + 0.863i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.590 - 0.807i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530690 + 0.269280i\)
\(L(\frac12)\) \(\approx\) \(0.530690 + 0.269280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.08 - 3.55i)T \)
43 \( 1 + (-4.74 + 4.52i)T \)
good2 \( 1 + (0.534 + 2.34i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-2.00 - 0.457i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.96 - 1.56i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 1.32iT - 7T^{2} \)
11 \( 1 + (0.0593 - 0.123i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.70 + 2.14i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (7.27 - 3.50i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.13 - 4.43i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (7.35 - 1.67i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-10.5 + 2.40i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 2.07iT - 37T^{2} \)
41 \( 1 + (-5.09 + 1.16i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (3.32 - 1.60i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.345 + 0.433i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.718 - 0.901i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (5.89 + 1.34i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (6.09 - 2.93i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-1.22 - 2.53i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.43 - 1.14i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.54iT - 79T^{2} \)
83 \( 1 + (2.48 - 10.8i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.333 - 1.46i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-3.02 + 6.29i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.50805885430183215230438966387, −9.815314636793296751534374835270, −8.923701583420191786050907892636, −8.289835240686649296422176827809, −7.61242768752000111309046205120, −6.04731541850274747226098954582, −4.23763789924513188217065233571, −3.69408015182936315742736115157, −2.78123547600274660104944208907, −2.01031948536797819583654522233, 0.28593041155748254942641955154, 2.49645464177916315314460182208, 4.27329093522944571239412052219, 4.64621212436721420251266024637, 6.18583884910864510091513387984, 7.08180005863853333904617535040, 7.71431076610939965765608115424, 8.428423127769119415597954192978, 8.891542697134275369846988554107, 9.637524735390780733266233335104

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