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Properties

Label 2-1161-9.7-c1-0-40
Degree $2$
Conductor $1161$
Sign $-0.632 - 0.774i$
Analytic cond. $9.27063$
Root an. cond. $3.04477$
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  + (1.31 − 2.28i)2-s + (−2.47 − 4.28i)4-s + (−1.39 − 2.42i)5-s + (1.99 − 3.44i)7-s − 7.78·8-s − 7.38·10-s + (2.07 − 3.58i)11-s + (1.53 + 2.65i)13-s + (−5.24 − 9.09i)14-s + (−5.30 + 9.19i)16-s + 5.89·17-s − 0.423·19-s + (−6.93 + 12.0i)20-s + (−5.46 − 9.46i)22-s + (−0.535 − 0.927i)23-s + ⋯
L(s)  = 1  + (0.932 − 1.61i)2-s + (−1.23 − 2.14i)4-s + (−0.626 − 1.08i)5-s + (0.752 − 1.30i)7-s − 2.75·8-s − 2.33·10-s + (0.624 − 1.08i)11-s + (0.425 + 0.736i)13-s + (−1.40 − 2.43i)14-s + (−1.32 + 2.29i)16-s + 1.42·17-s − 0.0971·19-s + (−1.55 + 2.68i)20-s + (−1.16 − 2.01i)22-s + (−0.111 − 0.193i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(9.27063\)
Root analytic conductor: \(3.04477\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1161,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.485038580\)
\(L(\frac12)\) \(\approx\) \(2.485038580\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.31 + 2.28i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.39 + 2.42i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.99 + 3.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.07 + 3.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.53 - 2.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 + 0.423T + 19T^{2} \)
23 \( 1 + (0.535 + 0.927i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.91 - 5.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.87 - 8.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + (-5.02 - 8.70i)T + (-20.5 + 35.5i)T^{2} \)
47 \( 1 + (4.13 - 7.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.04T + 53T^{2} \)
59 \( 1 + (1.40 + 2.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.49 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.766 + 1.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.85T + 71T^{2} \)
73 \( 1 - 4.68T + 73T^{2} \)
79 \( 1 + (5.77 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.655 - 1.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.630T + 89T^{2} \)
97 \( 1 + (2.05 - 3.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.512138459120914768414417284664, −8.640194076422546336695778465155, −7.88805999377295451384828970599, −6.47078368356895330549682217857, −5.30508277425033254284405545294, −4.57243826802931216246278860094, −3.93553709186041630496800775881, −3.21663632688334058415543106461, −1.32484605724073073909398141317, −0.993133351437661934353162837151, 2.50339018878787623462826351810, 3.61453327976629562919166546289, 4.40568819289647158977900750923, 5.57020702120030585653398976850, 5.93744488652174728217526775216, 7.02000988763262383007517450162, 7.68737031544272376250704601646, 8.163988529172884062770698747425, 9.146855595126231498740118229689, 10.17090385675750534566650567182

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