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Properties

Label 2-161-161.10-c1-0-4
Degree $2$
Conductor $161$
Sign $0.993 - 0.113i$
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  + (−0.0473 + 0.136i)2-s + (−0.509 − 0.988i)3-s + (1.55 + 1.22i)4-s + (0.991 − 0.0946i)5-s + (0.159 − 0.0229i)6-s + (−0.848 + 2.50i)7-s + (−0.484 + 0.311i)8-s + (1.02 − 1.43i)9-s + (−0.0340 + 0.140i)10-s + (3.11 − 1.07i)11-s + (0.416 − 2.16i)12-s + (−0.221 − 0.754i)13-s + (−0.302 − 0.234i)14-s + (−0.599 − 0.932i)15-s + (0.913 + 3.76i)16-s + (−4.01 − 1.60i)17-s + ⋯
L(s)  = 1  + (−0.0335 + 0.0967i)2-s + (−0.294 − 0.570i)3-s + (0.777 + 0.611i)4-s + (0.443 − 0.0423i)5-s + (0.0651 − 0.00936i)6-s + (−0.320 + 0.947i)7-s + (−0.171 + 0.110i)8-s + (0.340 − 0.478i)9-s + (−0.0107 + 0.0443i)10-s + (0.939 − 0.324i)11-s + (0.120 − 0.624i)12-s + (−0.0614 − 0.209i)13-s + (−0.0809 − 0.0627i)14-s + (−0.154 − 0.240i)15-s + (0.228 + 0.941i)16-s + (−0.973 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.993 - 0.113i$
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: \((2,\ 161,\ (\ :1/2),\ 0.993 - 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25890 + 0.0716294i\)
\(L(\frac12)\) \(\approx\) \(1.25890 + 0.0716294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.848 - 2.50i)T \)
23 \( 1 + (-2.00 - 4.35i)T \)
good2 \( 1 + (0.0473 - 0.136i)T + (-1.57 - 1.23i)T^{2} \)
3 \( 1 + (0.509 + 0.988i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-0.991 + 0.0946i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.11 + 1.07i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.221 + 0.754i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (4.01 + 1.60i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-2.45 + 0.981i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (1.03 + 7.19i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (8.24 - 0.392i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.700 + 0.498i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (5.91 + 2.70i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.83 - 9.08i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-7.88 + 4.54i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.344 - 0.361i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-4.80 - 1.16i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-6.36 - 3.28i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (1.70 + 8.83i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-2.35 - 2.72i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.82 - 3.59i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-6.45 + 6.77i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-0.611 - 1.33i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.196 - 4.12i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.10 + 2.42i)T + (-63.5 - 73.3i)T^{2} \)
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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

−12.81106988838569721816790670438, −11.77034022914202424867333713742, −11.43309751777573949384281234014, −9.670219077989410303400617719656, −8.838569643035937652101671076265, −7.42013377367887348272337250578, −6.53459086322136178050741679280, −5.69048794765243765326657137152, −3.56536701032249091031814343889, −1.99792438924193626969249662936, 1.79943094053870361341324054670, 3.86486225716852691157440366956, 5.20092552592745803182538411460, 6.53993864400136908608762457692, 7.26813679703019221032179598500, 9.137897788929918547093938410163, 10.11962598148585798584657019831, 10.67652805581955714500606654676, 11.57161889794238983163193576237, 12.86430020404690569930741399609

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