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Properties

Label 2-731-731.208-c1-0-63
Degree $2$
Conductor $731$
Sign $0.325 - 0.945i$
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  − 2.21i·2-s + (−0.0692 − 0.0185i)3-s − 2.91·4-s + (−1.25 − 0.336i)5-s + (−0.0410 + 0.153i)6-s + (0.0334 − 0.00896i)7-s + 2.01i·8-s + (−2.59 − 1.49i)9-s + (−0.746 + 2.78i)10-s + (1.16 + 1.16i)11-s + (0.201 + 0.0539i)12-s + (0.160 − 0.278i)13-s + (−0.0198 − 0.0741i)14-s + (0.0807 + 0.0465i)15-s − 1.34·16-s + (−4.09 − 0.450i)17-s + ⋯
L(s)  = 1  − 1.56i·2-s + (−0.0399 − 0.0107i)3-s − 1.45·4-s + (−0.561 − 0.150i)5-s + (−0.0167 + 0.0626i)6-s + (0.0126 − 0.00338i)7-s + 0.714i·8-s + (−0.864 − 0.499i)9-s + (−0.235 + 0.880i)10-s + (0.352 + 0.352i)11-s + (0.0581 + 0.0155i)12-s + (0.0445 − 0.0772i)13-s + (−0.00530 − 0.0198i)14-s + (0.0208 + 0.0120i)15-s − 0.336·16-s + (−0.994 − 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.325 - 0.945i$
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: \((2,\ 731,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125047 + 0.0892511i\)
\(L(\frac12)\) \(\approx\) \(0.125047 + 0.0892511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4.09 + 0.450i)T \)
43 \( 1 + (-5.85 + 2.95i)T \)
good2 \( 1 + 2.21iT - 2T^{2} \)
3 \( 1 + (0.0692 + 0.0185i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.25 + 0.336i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.0334 + 0.00896i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.16 - 1.16i)T + 11iT^{2} \)
13 \( 1 + (-0.160 + 0.278i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (1.99 - 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.55 - 5.80i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.16 - 4.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-5.50 - 1.47i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.17 - 0.313i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.15 + 8.15i)T + 41iT^{2} \)
47 \( 1 - 1.08T + 47T^{2} \)
53 \( 1 + (3.27 - 1.89i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.70iT - 59T^{2} \)
61 \( 1 + (0.925 - 0.248i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.73 + 9.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.20 + 11.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.890 - 3.32i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.35 - 1.97i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (12.5 - 7.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.98 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.65 + 7.65i)T - 97iT^{2} \)
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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

−9.866412107646085652450103832501, −9.034246066322715203689357409737, −8.402558016352592548980690992517, −7.12917975968717332942503330262, −6.00990274840862648988823591381, −4.66978989770550644558191155821, −3.83545802342528573081364218679, −2.93087612667671583709687065134, −1.68048009423099025025780079856, −0.07701501275878304879156110024, 2.56095548330231796082441120447, 4.14869075801791757891467640010, 4.92290036463625209455399731341, 6.12925101653366005765569362687, 6.52713854819554815547241531879, 7.66071304523333052941554254531, 8.350944383182459747124082978115, 8.820758084032051673039973796393, 10.03838957681902653602576217359, 11.26828296270028367559168279442

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