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Properties

Label 2-161-161.10-c1-0-10
Degree $2$
Conductor $161$
Sign $-0.267 + 0.963i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (0.220 − 0.635i)2-s + (−1.28 − 2.50i)3-s + (1.21 + 0.956i)4-s + (1.26 − 0.121i)5-s + (−1.87 + 0.269i)6-s + (−0.173 − 2.64i)7-s + (2.00 − 1.29i)8-s + (−2.85 + 4.00i)9-s + (0.202 − 0.833i)10-s + (−3.82 + 1.32i)11-s + (0.823 − 4.27i)12-s + (0.0588 + 0.200i)13-s + (−1.71 − 0.470i)14-s + (−1.93 − 3.01i)15-s + (0.350 + 1.44i)16-s + (−1.23 − 0.492i)17-s + ⋯
L(s)  = 1  + (0.155 − 0.449i)2-s + (−0.744 − 1.44i)3-s + (0.608 + 0.478i)4-s + (0.567 − 0.0541i)5-s + (−0.765 + 0.110i)6-s + (−0.0656 − 0.997i)7-s + (0.709 − 0.456i)8-s + (−0.950 + 1.33i)9-s + (0.0639 − 0.263i)10-s + (−1.15 + 0.398i)11-s + (0.237 − 1.23i)12-s + (0.0163 + 0.0556i)13-s + (−0.458 − 0.125i)14-s + (−0.500 − 0.779i)15-s + (0.0876 + 0.361i)16-s + (−0.298 − 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ -0.267 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712843 - 0.937895i\)
\(L(\frac12)\) \(\approx\) \(0.712843 - 0.937895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.173 + 2.64i)T \)
23 \( 1 + (-4.51 - 1.62i)T \)
good2 \( 1 + (-0.220 + 0.635i)T + (-1.57 - 1.23i)T^{2} \)
3 \( 1 + (1.28 + 2.50i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-1.26 + 0.121i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (3.82 - 1.32i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.0588 - 0.200i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.23 + 0.492i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-4.63 + 1.85i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.936 - 6.51i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-6.71 + 0.319i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-7.18 - 5.11i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (0.796 + 0.363i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.29 + 5.12i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.14 + 3.30i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (10.6 + 2.59i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-7.55 - 3.89i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (1.49 + 7.75i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-1.09 - 1.26i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.87 - 3.65i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (4.56 - 4.78i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (1.22 + 2.67i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.416 + 8.75i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (4.94 - 10.8i)T + (-63.5 - 73.3i)T^{2} \)
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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

−12.74671516065985520668848472989, −11.62882372109365595408126896297, −10.95876532833029979464874567068, −9.903814198954050883508005778512, −7.902985951099485073139033153635, −7.25145140573882685225731234868, −6.42157289671159398403867392815, −4.97275228286663669633825853571, −2.85460061943280084876538243063, −1.37410490302037935359014783229, 2.73471940313946769578474139341, 4.78034438634526479768259662612, 5.62252482763840662547926260674, 6.20499977177394169736808561145, 8.008451497130471420307572401703, 9.503543167204040744077775255302, 10.14358963366008646283812182591, 11.07835870382971216768147811892, 11.78828086991123696761243590506, 13.29708289369395344725519838423

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