Properties

Label 2-731-731.67-c1-0-7
Degree $2$
Conductor $731$
Sign $-0.711 + 0.702i$
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  + (−0.755 + 0.947i)2-s + (0.215 + 1.43i)3-s + (0.118 + 0.517i)4-s + (1.58 + 2.33i)5-s + (−1.51 − 0.877i)6-s + (−1.01 + 0.584i)7-s + (−2.76 − 1.33i)8-s + (0.863 − 0.266i)9-s + (−3.40 − 0.255i)10-s + (−4.33 − 0.989i)11-s + (−0.715 + 0.280i)12-s + (0.132 + 1.76i)13-s + (0.211 − 1.40i)14-s + (−2.99 + 2.77i)15-s + (2.39 − 1.15i)16-s + (−2.23 − 3.46i)17-s + ⋯
L(s)  = 1  + (−0.534 + 0.670i)2-s + (0.124 + 0.826i)3-s + (0.0590 + 0.258i)4-s + (0.710 + 1.04i)5-s + (−0.620 − 0.358i)6-s + (−0.382 + 0.221i)7-s + (−0.977 − 0.470i)8-s + (0.287 − 0.0887i)9-s + (−1.07 − 0.0808i)10-s + (−1.30 − 0.298i)11-s + (−0.206 + 0.0811i)12-s + (0.0367 + 0.490i)13-s + (0.0564 − 0.374i)14-s + (−0.773 + 0.717i)15-s + (0.598 − 0.288i)16-s + (−0.541 − 0.840i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332621 - 0.809858i\)
\(L(\frac12)\) \(\approx\) \(0.332621 - 0.809858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.23 + 3.46i)T \)
43 \( 1 + (-3.21 - 5.71i)T \)
good2 \( 1 + (0.755 - 0.947i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.215 - 1.43i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-1.58 - 2.33i)T + (-1.82 + 4.65i)T^{2} \)
7 \( 1 + (1.01 - 0.584i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.33 + 0.989i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.132 - 1.76i)T + (-12.8 + 1.93i)T^{2} \)
19 \( 1 + (3.67 + 1.13i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-1.03 + 1.11i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.31 - 8.69i)T + (-27.7 - 8.54i)T^{2} \)
31 \( 1 + (5.10 - 2.00i)T + (22.7 - 21.0i)T^{2} \)
37 \( 1 + (-8.49 - 4.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.60 - 2.87i)T + (9.12 + 39.9i)T^{2} \)
47 \( 1 + (2.29 + 10.0i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.241 - 3.21i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-9.16 + 4.41i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.01 - 1.57i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (7.09 + 2.18i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-4.06 - 4.37i)T + (-5.30 + 70.8i)T^{2} \)
73 \( 1 + (13.3 - 0.997i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (0.790 - 0.456i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.5 - 2.34i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (1.67 - 0.252i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-7.31 - 1.66i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.67482476482747251654229882005, −9.913129619276335714939384642729, −9.236984036171040746824266313191, −8.492648701175937601802472948144, −7.26297582945691955568781653016, −6.75832706001130333596755062288, −5.82725017147319815031261663274, −4.62573482775007302421672261588, −3.25401728933609471397948886735, −2.56090176791810162112870094253, 0.49683181150578865905427633522, 1.79987007657148121237734999797, 2.46369523268912641353278918557, 4.30264728943556131014286330801, 5.57728141612799831215525810250, 6.13679188063705124697075509237, 7.46045837130984182218387702986, 8.259552645912190493588645621470, 9.121237859876501644335682427481, 9.968406825586412575014274913770

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