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Properties

Label 2-731-17.16-c1-0-9
Degree $2$
Conductor $731$
Sign $-0.318 - 0.947i$
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  − 2.79·2-s + 1.46i·3-s + 5.80·4-s + 1.65i·5-s − 4.08i·6-s − 2.54i·7-s − 10.6·8-s + 0.862·9-s − 4.62i·10-s − 3.40i·11-s + 8.48i·12-s − 3.78·13-s + 7.10i·14-s − 2.42·15-s + 18.0·16-s + (1.31 + 3.90i)17-s + ⋯
L(s)  = 1  − 1.97·2-s + 0.844i·3-s + 2.90·4-s + 0.740i·5-s − 1.66i·6-s − 0.961i·7-s − 3.75·8-s + 0.287·9-s − 1.46i·10-s − 1.02i·11-s + 2.45i·12-s − 1.04·13-s + 1.90i·14-s − 0.625·15-s + 4.52·16-s + (0.318 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.318 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310366 + 0.431614i\)
\(L(\frac12)\) \(\approx\) \(0.310366 + 0.431614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.31 - 3.90i)T \)
43 \( 1 + T \)
good2 \( 1 + 2.79T + 2T^{2} \)
3 \( 1 - 1.46iT - 3T^{2} \)
5 \( 1 - 1.65iT - 5T^{2} \)
7 \( 1 + 2.54iT - 7T^{2} \)
11 \( 1 + 3.40iT - 11T^{2} \)
13 \( 1 + 3.78T + 13T^{2} \)
19 \( 1 + 0.391T + 19T^{2} \)
23 \( 1 - 6.71iT - 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 + 4.70iT - 31T^{2} \)
37 \( 1 - 6.05iT - 37T^{2} \)
41 \( 1 - 4.77iT - 41T^{2} \)
47 \( 1 - 3.54T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 4.63iT - 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 - 5.85iT - 73T^{2} \)
79 \( 1 - 14.7iT - 79T^{2} \)
83 \( 1 + 7.29T + 83T^{2} \)
89 \( 1 + 5.50T + 89T^{2} \)
97 \( 1 - 1.97iT - 97T^{2} \)
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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

−10.29164213643441172478087546705, −9.991481330281237885766977230760, −9.137841663946969909820267262724, −8.149410879044672919340455653099, −7.36082629967872002776017166653, −6.75864283125810922891880239180, −5.61639873507513685095888130134, −3.80357938770861011013790846563, −2.84729494875018613673278443879, −1.24855316688691524849776988161, 0.56784369690517976506254980003, 1.93236212774722709542546510362, 2.57547276184169494747958074296, 4.93064562065928345180633396408, 6.18210081427729278029686750492, 7.14969603652445563937815471120, 7.53775078513423136771217673661, 8.571303074154150638059063881971, 9.151378421883567954872276502249, 9.877274650112859387862115384557

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