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Properties

Label 2-731-17.2-c1-0-45
Degree $2$
Conductor $731$
Sign $-0.955 + 0.293i$
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  + (−0.0668 − 0.0668i)2-s + (−2.64 + 1.09i)3-s − 1.99i·4-s + (−0.846 − 2.04i)5-s + (0.249 + 0.103i)6-s + (1.54 − 3.72i)7-s + (−0.266 + 0.266i)8-s + (3.65 − 3.65i)9-s + (−0.0800 + 0.193i)10-s + (2.20 + 0.915i)11-s + (2.17 + 5.25i)12-s − 0.0553i·13-s + (−0.351 + 0.145i)14-s + (4.47 + 4.47i)15-s − 3.94·16-s + (3.88 − 1.39i)17-s + ⋯
L(s)  = 1  + (−0.0472 − 0.0472i)2-s + (−1.52 + 0.631i)3-s − 0.995i·4-s + (−0.378 − 0.913i)5-s + (0.101 + 0.0422i)6-s + (0.582 − 1.40i)7-s + (−0.0943 + 0.0943i)8-s + (1.21 − 1.21i)9-s + (−0.0253 + 0.0610i)10-s + (0.666 + 0.276i)11-s + (0.628 + 1.51i)12-s − 0.0153i·13-s + (−0.0940 + 0.0389i)14-s + (1.15 + 1.15i)15-s − 0.986·16-s + (0.941 − 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0953138 - 0.634777i\)
\(L(\frac12)\) \(\approx\) \(0.0953138 - 0.634777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.88 + 1.39i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.0668 + 0.0668i)T + 2iT^{2} \)
3 \( 1 + (2.64 - 1.09i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.846 + 2.04i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.54 + 3.72i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.20 - 0.915i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.0553iT - 13T^{2} \)
19 \( 1 + (4.14 + 4.14i)T + 19iT^{2} \)
23 \( 1 + (2.47 + 1.02i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.65 - 4.00i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.40 + 1.82i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (7.74 - 3.21i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.0490 + 0.118i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 + 5.01iT - 47T^{2} \)
53 \( 1 + (3.29 + 3.29i)T + 53iT^{2} \)
59 \( 1 + (-6.50 + 6.50i)T - 59iT^{2} \)
61 \( 1 + (2.12 - 5.13i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + (5.20 - 2.15i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.57 - 3.79i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-8.63 - 3.57i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.93 + 6.93i)T + 83iT^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 + (-1.07 - 2.59i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.38136300093419614817828501221, −9.465549990333374532198263402395, −8.402350775430766573734804529350, −7.04955895177854870493955094559, −6.38392097485550734369504147734, −5.19598654651424967046214054588, −4.70611658945066644575616405563, −4.05396613243759710372893676447, −1.31117661300793905869895525085, −0.46216893655286801126514372331, 1.84718408676739470096860089427, 3.23094653271361302691489619242, 4.49299182871999742566839500367, 5.77724892385299639647792723798, 6.26511494110424362698365380962, 7.22446294617992690970963585806, 8.022664299748000903823646936106, 8.813301301043631919799609872913, 10.24000644372234090783291330453, 11.10163975704420003859198022021

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