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Properties

Label 2-161-161.68-c1-0-13
Degree $2$
Conductor $161$
Sign $-0.317 + 0.948i$
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  + (1.14 − 1.99i)2-s + (0.604 − 0.349i)3-s + (−1.64 − 2.84i)4-s + (0.630 − 1.09i)5-s − 1.60i·6-s + (0.738 + 2.54i)7-s − 2.94·8-s + (−1.25 + 2.17i)9-s + (−1.44 − 2.51i)10-s + (−3.12 + 1.80i)11-s + (−1.98 − 1.14i)12-s + 1.83i·13-s + (5.90 + 1.45i)14-s − 0.880i·15-s + (−0.106 + 0.185i)16-s + (−3.52 − 6.09i)17-s + ⋯
L(s)  = 1  + (0.812 − 1.40i)2-s + (0.349 − 0.201i)3-s + (−0.820 − 1.42i)4-s + (0.282 − 0.488i)5-s − 0.655i·6-s + (0.278 + 0.960i)7-s − 1.04·8-s + (−0.418 + 0.725i)9-s + (−0.458 − 0.794i)10-s + (−0.941 + 0.543i)11-s + (−0.573 − 0.330i)12-s + 0.509i·13-s + (1.57 + 0.387i)14-s − 0.227i·15-s + (−0.0267 + 0.0463i)16-s + (−0.854 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.317 + 0.948i$
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: \((2,\ 161,\ (\ :1/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04277 - 1.44873i\)
\(L(\frac12)\) \(\approx\) \(1.04277 - 1.44873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.738 - 2.54i)T \)
23 \( 1 + (-4.25 - 2.21i)T \)
good2 \( 1 + (-1.14 + 1.99i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.604 + 0.349i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.630 + 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.12 - 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.83iT - 13T^{2} \)
17 \( 1 + (3.52 + 6.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 2.13i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 + (-0.372 + 0.215i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.54 - 2.04i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.63iT - 41T^{2} \)
43 \( 1 - 0.231iT - 43T^{2} \)
47 \( 1 + (3.81 + 2.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.46 - 3.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.28 + 1.89i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.88 + 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 1.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.24T + 71T^{2} \)
73 \( 1 + (13.5 - 7.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 + 3.40i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 + (6.20 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.25T + 97T^{2} \)
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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

−12.71512549711008933720766213346, −11.53888889033073311420298045998, −11.05576821776993990140758609256, −9.613927081846198533699140767632, −8.871879257310536212765486250944, −7.39180842701811105568860852260, −5.28622298045511918955361364531, −4.87101621508096310726906976792, −2.87403912951152061044107876515, −2.02345852592123586836323209516, 3.25566573161341272840863712189, 4.44389373751923019169885027645, 5.82488948848426711859410203859, 6.64198062836667678605689492105, 7.83286478251042627621106007180, 8.572086941738229754202409145632, 10.20294113678944288913275830015, 11.07875402457912474187082472988, 12.86761794098738590062861959215, 13.35688081493407244851872731391

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