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Properties

Label 2-731-731.208-c1-0-2
Degree $2$
Conductor $731$
Sign $0.918 - 0.395i$
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  − 2.76i·2-s + (1.78 + 0.479i)3-s − 5.62·4-s + (−4.22 − 1.13i)5-s + (1.32 − 4.94i)6-s + (−0.560 + 0.150i)7-s + 10.0i·8-s + (0.375 + 0.216i)9-s + (−3.13 + 11.6i)10-s + (0.496 + 0.496i)11-s + (−10.0 − 2.69i)12-s + (1.00 − 1.74i)13-s + (0.414 + 1.54i)14-s + (−7.02 − 4.05i)15-s + 16.4·16-s + (4.04 + 0.818i)17-s + ⋯
L(s)  = 1  − 1.95i·2-s + (1.03 + 0.276i)3-s − 2.81·4-s + (−1.89 − 0.506i)5-s + (0.540 − 2.01i)6-s + (−0.211 + 0.0567i)7-s + 3.54i·8-s + (0.125 + 0.0721i)9-s + (−0.989 + 3.69i)10-s + (0.149 + 0.149i)11-s + (−2.90 − 0.779i)12-s + (0.279 − 0.484i)13-s + (0.110 + 0.413i)14-s + (−1.81 − 1.04i)15-s + 4.10·16-s + (0.980 + 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.918 - 0.395i$
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: \((2,\ 731,\ (\ :1/2),\ 0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207431 + 0.0427534i\)
\(L(\frac12)\) \(\approx\) \(0.207431 + 0.0427534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.04 - 0.818i)T \)
43 \( 1 + (6.31 + 1.78i)T \)
good2 \( 1 + 2.76iT - 2T^{2} \)
3 \( 1 + (-1.78 - 0.479i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (4.22 + 1.13i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.560 - 0.150i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.496 - 0.496i)T + 11iT^{2} \)
13 \( 1 + (-1.00 + 1.74i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (2.90 - 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.15 - 8.05i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.14 - 4.26i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (6.88 + 1.84i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.64 + 0.441i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.356 + 0.356i)T + 41iT^{2} \)
47 \( 1 + 4.18T + 47T^{2} \)
53 \( 1 + (9.50 - 5.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.07iT - 59T^{2} \)
61 \( 1 + (-2.50 + 0.670i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.209 - 0.363i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.789 - 2.94i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.73 + 6.46i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.67 - 0.984i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-5.82 + 3.36i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.955 + 1.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.957 - 0.957i)T - 97iT^{2} \)
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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

−10.58204558407261785075431112870, −9.581127552089831543903121359737, −8.949529999299276016446767499274, −8.164871502717745890488447435489, −7.73527419292204805471384991644, −5.37280016752958873678849414719, −4.26902248836937170314138467930, −3.44322160008104255950870359987, −3.28904801474360614037133908948, −1.52951892387126077622884988914, 0.10313288363552424787503335388, 3.24745696500186353752164471516, 3.96716346325865752683509285035, 4.88379564161746444601310715645, 6.39508807165948865717157224197, 6.98839791107777564493497182184, 7.80368951971552808862377454364, 8.308952976588488722637779240767, 8.757248167600767429023682390892, 9.899979992676372103166538711702

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