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Properties

Label 2-161-161.68-c1-0-6
Degree $2$
Conductor $161$
Sign $0.507 - 0.861i$
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  + (−0.559 + 0.969i)2-s + (2.13 − 1.23i)3-s + (0.372 + 0.645i)4-s + (−0.871 + 1.50i)5-s + 2.75i·6-s + (−0.785 + 2.52i)7-s − 3.07·8-s + (1.53 − 2.66i)9-s + (−0.975 − 1.68i)10-s + (4.42 − 2.55i)11-s + (1.59 + 0.918i)12-s − 1.11i·13-s + (−2.01 − 2.17i)14-s + 4.29i·15-s + (0.976 − 1.69i)16-s + (−1.62 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.395 + 0.685i)2-s + (1.23 − 0.711i)3-s + (0.186 + 0.322i)4-s + (−0.389 + 0.674i)5-s + 1.12i·6-s + (−0.297 + 0.954i)7-s − 1.08·8-s + (0.512 − 0.886i)9-s + (−0.308 − 0.534i)10-s + (1.33 − 0.769i)11-s + (0.459 + 0.265i)12-s − 0.309i·13-s + (−0.537 − 0.581i)14-s + 1.10i·15-s + (0.244 − 0.422i)16-s + (−0.394 − 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ 0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13218 + 0.646812i\)
\(L(\frac12)\) \(\approx\) \(1.13218 + 0.646812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.785 - 2.52i)T \)
23 \( 1 + (4.76 + 0.499i)T \)
good2 \( 1 + (0.559 - 0.969i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.13 + 1.23i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.871 - 1.50i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.42 + 2.55i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.74 + 4.74i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 + (-4.15 + 2.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.07 - 0.619i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 7.76iT - 43T^{2} \)
47 \( 1 + (-1.68 - 0.972i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.912 + 0.526i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.93 - 4.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.59 - 6.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.28 - 4.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 + (-5.01 + 2.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.52 + 5.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (5.34 - 9.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−13.21992417828459527624487356088, −12.02767006298554321315826615047, −11.35200313474712428030918812690, −9.253028284810469308877593937524, −8.883805978456413034008651916188, −7.77883520472449609029984148346, −7.01827096021616567630983081866, −6.04739187518912033738077173784, −3.46459840327484294226962780875, −2.59304067524168429186896821376, 1.66966884574055433533808159256, 3.54834700039656961383275738515, 4.33383094682761684975757504276, 6.39892800750024859367845069547, 7.88107746808528651599700347516, 8.982507083428816010347735882342, 9.680779034713982077274554937441, 10.34228528825902482635318138498, 11.66005720790818311120169645053, 12.52521317561719268387565687134

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