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Properties

Label 2-731-731.208-c1-0-62
Degree $2$
Conductor $731$
Sign $0.572 - 0.819i$
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  − 2.67i·2-s + (−0.393 − 0.105i)3-s − 5.17·4-s + (3.35 + 0.900i)5-s + (−0.282 + 1.05i)6-s + (−1.82 + 0.489i)7-s + 8.50i·8-s + (−2.45 − 1.41i)9-s + (2.41 − 8.99i)10-s + (−2.86 − 2.86i)11-s + (2.03 + 0.545i)12-s + (−2.42 + 4.20i)13-s + (1.31 + 4.89i)14-s + (−1.22 − 0.707i)15-s + 12.4·16-s + (−0.418 − 4.10i)17-s + ⋯
L(s)  = 1  − 1.89i·2-s + (−0.226 − 0.0608i)3-s − 2.58·4-s + (1.50 + 0.402i)5-s + (−0.115 + 0.429i)6-s + (−0.690 + 0.185i)7-s + 3.00i·8-s + (−0.818 − 0.472i)9-s + (0.762 − 2.84i)10-s + (−0.863 − 0.863i)11-s + (0.587 + 0.157i)12-s + (−0.673 + 1.16i)13-s + (0.350 + 1.30i)14-s + (−0.316 − 0.182i)15-s + 3.10·16-s + (−0.101 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.572 - 0.819i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.572 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0173363 + 0.00903982i\)
\(L(\frac12)\) \(\approx\) \(0.0173363 + 0.00903982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.418 + 4.10i)T \)
43 \( 1 + (-0.144 + 6.55i)T \)
good2 \( 1 + 2.67iT - 2T^{2} \)
3 \( 1 + (0.393 + 0.105i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.35 - 0.900i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.82 - 0.489i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.86 + 2.86i)T + 11iT^{2} \)
13 \( 1 + (2.42 - 4.20i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (5.21 - 3.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.191 - 0.715i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.53 - 9.44i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (6.00 + 1.60i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (9.45 + 2.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.08 - 4.08i)T + 41iT^{2} \)
47 \( 1 - 1.64T + 47T^{2} \)
53 \( 1 + (-8.63 + 4.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 + (4.75 - 1.27i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.04 - 1.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.951 + 3.54i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.53 + 5.74i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.86 - 0.499i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (4.88 - 2.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.557 + 0.966i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.87 - 6.87i)T - 97iT^{2} \)
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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

−9.849628053921855840055249097968, −9.162542722031748472351101401967, −8.667080386448649908847103947336, −6.80047090072153860089140210077, −5.74183211452159262845268855760, −5.05914305978202936384512477356, −3.50483742704285279628087716208, −2.68576817564496248097164305446, −1.88225138698810952548331463803, −0.009561792308506420963411087264, 2.49719444902164815605025131456, 4.42104993552287947529737114123, 5.33042939461873907511774327813, 5.80041947119645051421765652032, 6.54838506882375672906344373376, 7.55143887534601925891918319793, 8.426897283304617195698970011194, 9.135253711316780308495206211459, 10.17810136751502449366944202512, 10.37845076505698417461987315249

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