Properties

Label 2-731-17.2-c1-0-14
Degree $2$
Conductor $731$
Sign $0.981 + 0.192i$
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  + (−1.56 − 1.56i)2-s + (−2.36 + 0.978i)3-s + 2.87i·4-s + (1.21 + 2.92i)5-s + (5.21 + 2.16i)6-s + (1.72 − 4.16i)7-s + (1.36 − 1.36i)8-s + (2.50 − 2.50i)9-s + (2.67 − 6.45i)10-s + (2.87 + 1.18i)11-s + (−2.81 − 6.79i)12-s + 1.03i·13-s + (−9.18 + 3.80i)14-s + (−5.72 − 5.72i)15-s + 1.48·16-s + (−4.07 − 0.604i)17-s + ⋯
L(s)  = 1  + (−1.10 − 1.10i)2-s + (−1.36 + 0.565i)3-s + 1.43i·4-s + (0.541 + 1.30i)5-s + (2.13 + 0.882i)6-s + (0.651 − 1.57i)7-s + (0.483 − 0.483i)8-s + (0.834 − 0.834i)9-s + (0.846 − 2.04i)10-s + (0.866 + 0.358i)11-s + (−0.812 − 1.96i)12-s + 0.286i·13-s + (−2.45 + 1.01i)14-s + (−1.47 − 1.47i)15-s + 0.370·16-s + (−0.989 − 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.981 + 0.192i$
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: \((2,\ 731,\ (\ :1/2),\ 0.981 + 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583621 - 0.0566741i\)
\(L(\frac12)\) \(\approx\) \(0.583621 - 0.0566741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4.07 + 0.604i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.56 + 1.56i)T + 2iT^{2} \)
3 \( 1 + (2.36 - 0.978i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.21 - 2.92i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.72 + 4.16i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.87 - 1.18i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.03iT - 13T^{2} \)
19 \( 1 + (0.211 + 0.211i)T + 19iT^{2} \)
23 \( 1 + (3.32 + 1.37i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.75 - 4.22i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.62 + 1.91i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-6.56 + 2.71i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.389 + 0.939i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 3.37iT - 47T^{2} \)
53 \( 1 + (-9.44 - 9.44i)T + 53iT^{2} \)
59 \( 1 + (-5.23 + 5.23i)T - 59iT^{2} \)
61 \( 1 + (2.69 - 6.51i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 7.74T + 67T^{2} \)
71 \( 1 + (-8.36 + 3.46i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.57 + 11.0i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-13.6 - 5.67i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-9.21 - 9.21i)T + 83iT^{2} \)
89 \( 1 - 0.648iT - 89T^{2} \)
97 \( 1 + (-3.19 - 7.70i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.56030034581897110415658287394, −9.989050254968862701717106287102, −9.155495266936318138954217132596, −7.77055461252633587527332925934, −6.83827145307360272789549159912, −6.17993845523107397889215672677, −4.62250893708883721551827283121, −3.81912485918004694738366735012, −2.28615210217402202378365712726, −0.922240625515251551603819249486, 0.75086102319062643586805584499, 1.87398816389126226272517413769, 4.69622910819946060820927901974, 5.55658287205530699544077774350, 6.01617290072242174430960071854, 6.68651247836376682703111535806, 8.049223797406660014756569624092, 8.627273164011327297154367934138, 9.206381004957035233788090317736, 10.12100700910694676302069100603

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