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Properties

Label 2-731-731.208-c1-0-55
Degree $2$
Conductor $731$
Sign $-0.995 - 0.0932i$
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  − 2.26i·2-s + (1.08 + 0.289i)3-s − 3.15·4-s + (1.08 + 0.289i)5-s + (0.657 − 2.45i)6-s + (−0.970 + 0.259i)7-s + 2.61i·8-s + (−1.51 − 0.873i)9-s + (0.657 − 2.45i)10-s + (−3.42 − 3.42i)11-s + (−3.40 − 0.912i)12-s + (1.79 − 3.11i)13-s + (0.589 + 2.20i)14-s + (1.08 + 0.626i)15-s − 0.376·16-s + (3.78 + 1.63i)17-s + ⋯
L(s)  = 1  − 1.60i·2-s + (0.624 + 0.167i)3-s − 1.57·4-s + (0.483 + 0.129i)5-s + (0.268 − 1.00i)6-s + (−0.366 + 0.0982i)7-s + 0.922i·8-s + (−0.504 − 0.291i)9-s + (0.207 − 0.775i)10-s + (−1.03 − 1.03i)11-s + (−0.983 − 0.263i)12-s + (0.498 − 0.863i)13-s + (0.157 + 0.588i)14-s + (0.280 + 0.161i)15-s − 0.0942·16-s + (0.918 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.995 - 0.0932i$
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: \((2,\ 731,\ (\ :1/2),\ -0.995 - 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0674113 + 1.44262i\)
\(L(\frac12)\) \(\approx\) \(0.0674113 + 1.44262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.78 - 1.63i)T \)
43 \( 1 + (-5.53 - 3.52i)T \)
good2 \( 1 + 2.26iT - 2T^{2} \)
3 \( 1 + (-1.08 - 0.289i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.08 - 0.289i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.970 - 0.259i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.42 + 3.42i)T + 11iT^{2} \)
13 \( 1 + (-1.79 + 3.11i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-4.38 + 2.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.419 - 1.56i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.30 + 4.87i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.69 + 0.453i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.945 + 0.253i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.91 + 3.91i)T + 41iT^{2} \)
47 \( 1 - 7.97T + 47T^{2} \)
53 \( 1 + (7.88 - 4.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.1iT - 59T^{2} \)
61 \( 1 + (-5.55 + 1.48i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.49 - 7.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.85 - 6.91i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.65 - 13.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.7 + 3.14i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-9.37 + 5.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.51 - 7.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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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

−9.925097101892262642512484188284, −9.504353875565495791666194685367, −8.503751575979727694361085018784, −7.78767879122824062086551836823, −6.04952573139853698742305130226, −5.35307149705971876462727807942, −3.69120736161373481415474209513, −3.14723829815328800529324219555, −2.35618902360684178869441595027, −0.67712751706575571265275911511, 2.05330275851599805742994888801, 3.49121997105104732698337521157, 4.97928997244702112482317524748, 5.53934653940646433664685609179, 6.54362280500150582161435019293, 7.56442366039107961857109470118, 7.84628264771669213688524256868, 9.009241778604430681685786758710, 9.496649836191263777641350858112, 10.50146575451047192127547037553

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