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Properties

Label 2-731-731.208-c1-0-36
Degree $2$
Conductor $731$
Sign $-0.297 + 0.954i$
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  − 2.61i·2-s + (2.69 + 0.722i)3-s − 4.81·4-s + (1.96 + 0.525i)5-s + (1.88 − 7.03i)6-s + (−1.40 + 0.377i)7-s + 7.34i·8-s + (4.14 + 2.39i)9-s + (1.37 − 5.11i)10-s + (2.06 + 2.06i)11-s + (−12.9 − 3.47i)12-s + (2.11 − 3.67i)13-s + (0.984 + 3.67i)14-s + (4.90 + 2.83i)15-s + 9.54·16-s + (−3.77 − 1.65i)17-s + ⋯
L(s)  = 1  − 1.84i·2-s + (1.55 + 0.417i)3-s − 2.40·4-s + (0.876 + 0.234i)5-s + (0.769 − 2.87i)6-s + (−0.531 + 0.142i)7-s + 2.59i·8-s + (1.38 + 0.798i)9-s + (0.433 − 1.61i)10-s + (0.623 + 0.623i)11-s + (−3.74 − 1.00i)12-s + (0.587 − 1.01i)13-s + (0.263 + 0.981i)14-s + (1.26 + 0.731i)15-s + 2.38·16-s + (−0.916 − 0.400i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.297 + 0.954i$
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: \((2,\ 731,\ (\ :1/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48676 - 2.02098i\)
\(L(\frac12)\) \(\approx\) \(1.48676 - 2.02098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.77 + 1.65i)T \)
43 \( 1 + (4.57 - 4.70i)T \)
good2 \( 1 + 2.61iT - 2T^{2} \)
3 \( 1 + (-2.69 - 0.722i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.96 - 0.525i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.40 - 0.377i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.06 - 2.06i)T + 11iT^{2} \)
13 \( 1 + (-2.11 + 3.67i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-5.27 + 3.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.69 + 6.32i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.41 - 9.00i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.304 + 0.0815i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.80 - 1.82i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.58 + 1.58i)T + 41iT^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 + (-2.38 + 1.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.10iT - 59T^{2} \)
61 \( 1 + (10.1 - 2.71i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.64 + 2.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0110 + 0.0411i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.935 - 3.49i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.71 - 2.33i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (8.30 - 4.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.49 - 6.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.86 + 4.86i)T - 97iT^{2} \)
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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

−10.11617853679889473666421905083, −9.292941764035559056534605320375, −9.100644042468602919519901020778, −8.111240245237505837492258697949, −6.66858346470674702632024940947, −5.06226723652499527081610815261, −4.10712362404057596974132484962, −2.93197819737746732222284991123, −2.74765244195916493272060040109, −1.42497150054485613443359590527, 1.61359520634698658128832251035, 3.39667539685999631544401522119, 4.26419876520196821376189637591, 5.74500019627533438349612019023, 6.37552114429504577883384760268, 7.19270785073875792344455582423, 8.042927719102645568294524051228, 8.730969950825640799531498017857, 9.496079759944417267877769897393, 9.668282441792805812613671614583

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