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Properties

Label 2-731-731.208-c1-0-39
Degree $2$
Conductor $731$
Sign $-0.998 - 0.0548i$
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  − 1.44i·2-s + (−2.41 − 0.647i)3-s − 0.0931·4-s + (0.138 + 0.0371i)5-s + (−0.937 + 3.49i)6-s + (0.626 − 0.167i)7-s − 2.75i·8-s + (2.82 + 1.63i)9-s + (0.0537 − 0.200i)10-s + (2.06 + 2.06i)11-s + (0.225 + 0.0603i)12-s + (2.14 − 3.72i)13-s + (−0.242 − 0.905i)14-s + (−0.311 − 0.179i)15-s − 4.17·16-s + (−1.99 − 3.60i)17-s + ⋯
L(s)  = 1  − 1.02i·2-s + (−1.39 − 0.373i)3-s − 0.0465·4-s + (0.0620 + 0.0166i)5-s + (−0.382 + 1.42i)6-s + (0.236 − 0.0634i)7-s − 0.975i·8-s + (0.941 + 0.543i)9-s + (0.0169 − 0.0634i)10-s + (0.623 + 0.623i)11-s + (0.0649 + 0.0174i)12-s + (0.595 − 1.03i)13-s + (−0.0648 − 0.242i)14-s + (−0.0803 − 0.0463i)15-s − 1.04·16-s + (−0.484 − 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.998 - 0.0548i$
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: \((2,\ 731,\ (\ :1/2),\ -0.998 - 0.0548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0248380 + 0.904316i\)
\(L(\frac12)\) \(\approx\) \(0.0248380 + 0.904316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.99 + 3.60i)T \)
43 \( 1 + (6.50 - 0.839i)T \)
good2 \( 1 + 1.44iT - 2T^{2} \)
3 \( 1 + (2.41 + 0.647i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.138 - 0.0371i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.626 + 0.167i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.06 - 2.06i)T + 11iT^{2} \)
13 \( 1 + (-2.14 + 3.72i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-0.179 + 0.103i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.70 + 6.36i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.54 - 5.77i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.69 - 0.990i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (10.5 + 2.83i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.367 - 0.367i)T + 41iT^{2} \)
47 \( 1 - 1.28T + 47T^{2} \)
53 \( 1 + (2.11 - 1.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.5iT - 59T^{2} \)
61 \( 1 + (-4.26 + 1.14i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.99 - 5.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.26 + 8.45i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.42 - 12.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.12 - 0.300i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-10.3 + 5.94i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.78 + 3.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.89 - 7.89i)T - 97iT^{2} \)
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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

−10.42822108734758140660607094844, −9.483311106338719245199384170139, −8.275247596115369326982718175293, −6.87093598570169425221615557355, −6.60603957514270379681237582918, −5.37012912417974077884196246364, −4.46170625113333686559820082305, −3.13601931212202345990379223483, −1.74427142124236967554800095012, −0.56919210082927219018119740186, 1.67860543204006184550966133559, 3.77858576078131185126268431753, 4.86031231819888302849741975823, 5.74756527070994627850861947468, 6.29992882049217727945877093816, 6.96200130093496014015362913124, 8.155048141627963504961185226277, 8.948999923603923513165537203840, 10.06104209219398823088193581100, 11.07409288411786219328047984743

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