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Properties

Label 2-731-731.135-c1-0-32
Degree $2$
Conductor $731$
Sign $-0.702 + 0.711i$
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  − 2.51·2-s + (1.76 + 1.01i)3-s + 4.30·4-s + (−1.18 − 0.684i)5-s + (−4.42 − 2.55i)6-s + (−3.84 + 2.21i)7-s − 5.79·8-s + (0.567 + 0.983i)9-s + (2.97 + 1.72i)10-s + 5.71i·11-s + (7.58 + 4.37i)12-s + (−0.967 − 1.67i)13-s + (9.65 − 5.57i)14-s + (−1.39 − 2.41i)15-s + 5.93·16-s + (1.23 + 3.93i)17-s + ⋯
L(s)  = 1  − 1.77·2-s + (1.01 + 0.587i)3-s + 2.15·4-s + (−0.530 − 0.306i)5-s + (−1.80 − 1.04i)6-s + (−1.45 + 0.838i)7-s − 2.04·8-s + (0.189 + 0.327i)9-s + (0.942 + 0.543i)10-s + 1.72i·11-s + (2.18 + 1.26i)12-s + (−0.268 − 0.464i)13-s + (2.57 − 1.48i)14-s + (−0.359 − 0.622i)15-s + 1.48·16-s + (0.299 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000736931 - 0.00176288i\)
\(L(\frac12)\) \(\approx\) \(0.000736931 - 0.00176288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.23 - 3.93i)T \)
43 \( 1 + (-4.98 + 4.25i)T \)
good2 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 + (-1.76 - 1.01i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.18 + 0.684i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.84 - 2.21i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.71iT - 11T^{2} \)
13 \( 1 + (0.967 + 1.67i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-3.28 + 5.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.56 + 2.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.60 + 0.925i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.10 + 4.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.467 - 0.269i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.51iT - 41T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (2.08 - 3.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.09T + 59T^{2} \)
61 \( 1 + (-1.58 + 0.917i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.10 - 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.83 - 2.79i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.46 - 3.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.6 - 6.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.69 - 8.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.13 + 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.8iT - 97T^{2} \)
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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

−9.978685157220761945479927134665, −9.174930105904024991251698109624, −8.691255646621895063330513090638, −7.75706068148256228270311501505, −7.05093498669088672652782166273, −5.95841677212399254971226632400, −4.24940316902713779600175847747, −2.99227019656696279779039641774, −2.17406657456065827764883367972, −0.00151079920365868248984230885, 1.44907810740163173645898376689, 3.12488501992871830416353318132, 3.38373002088769308786411894833, 5.96384407169954916165452859905, 6.88605599314116793994645376217, 7.62720985298439930429172768489, 8.001915502989045331822914306197, 9.096121944389108446425476340547, 9.542329548148185478735375662466, 10.43755025741695144042833197294

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