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Properties

Label 2-161-161.160-c3-0-32
Degree $2$
Conductor $161$
Sign $0.983 - 0.181i$
Analytic cond. $9.49930$
Root an. cond. $3.08209$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s + 18.5i·7-s + 45·8-s + 27·9-s − 26.4i·11-s + 92.6i·14-s + 89·16-s + 135·18-s − 132. i·22-s + (−20 − 108. i)23-s − 125·25-s + 314. i·28-s − 166·29-s + 85·32-s + ⋯
L(s)  = 1  + 1.76·2-s + 2.12·4-s + 0.999i·7-s + 1.98·8-s + 9-s − 0.725i·11-s + 1.76i·14-s + 1.39·16-s + 1.76·18-s − 1.28i·22-s + (−0.181 − 0.983i)23-s − 25-s + 2.12i·28-s − 1.06·29-s + 0.469·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ 0.983 - 0.181i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.69295 + 0.429011i\)
\(L(\frac12)\) \(\approx\) \(4.69295 + 0.429011i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 18.5iT \)
23 \( 1 + (20 + 108. i)T \)
good2 \( 1 - 5T + 8T^{2} \)
3 \( 1 - 27T^{2} \)
5 \( 1 + 125T^{2} \)
11 \( 1 + 26.4iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
29 \( 1 + 166T + 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + 10.5iT - 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 534. iT - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 + 497. iT - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 + 2.26e5T^{2} \)
67 \( 1 + 809. iT - 3.00e5T^{2} \)
71 \( 1 - 688T + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 238. iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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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

−12.66483415066410089430185407408, −11.79069305809781801682345470132, −10.92749576957603219408250581208, −9.528759561709270091067446313542, −7.997999759952756044890606199674, −6.62083750291022937709429370568, −5.76509989011984259122926465998, −4.66934068274913375718029489499, −3.47418187918152474590316900370, −2.09740496270781321880819806280, 1.83335910483338540894471733642, 3.67925210366449748770050785583, 4.38767252079174111000164465038, 5.60715643180661942277553090779, 6.96079375010028624354289122876, 7.53316939992615302929643224494, 9.692225784698898351787711010144, 10.67527403080039798635861884268, 11.73379302900258344856035203407, 12.65311215866959000841276178568

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