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Properties

Label 2-162-81.13-c1-0-7
Degree $2$
Conductor $162$
Sign $-0.449 + 0.893i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
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  + (−0.893 − 0.448i)2-s + (0.647 − 1.60i)3-s + (0.597 + 0.802i)4-s + (−0.750 − 2.50i)5-s + (−1.29 + 1.14i)6-s + (0.318 + 0.738i)7-s + (−0.173 − 0.984i)8-s + (−2.16 − 2.08i)9-s + (−0.454 + 2.57i)10-s + (0.151 − 0.0358i)11-s + (1.67 − 0.439i)12-s + (−0.574 − 0.377i)13-s + (0.0467 − 0.802i)14-s + (−4.51 − 0.417i)15-s + (−0.286 + 0.957i)16-s + (0.0626 − 0.0228i)17-s + ⋯
L(s)  = 1  + (−0.631 − 0.317i)2-s + (0.373 − 0.927i)3-s + (0.298 + 0.401i)4-s + (−0.335 − 1.12i)5-s + (−0.530 + 0.467i)6-s + (0.120 + 0.279i)7-s + (−0.0613 − 0.348i)8-s + (−0.720 − 0.693i)9-s + (−0.143 + 0.814i)10-s + (0.0456 − 0.0108i)11-s + (0.483 − 0.126i)12-s + (−0.159 − 0.104i)13-s + (0.0124 − 0.214i)14-s + (−1.16 − 0.107i)15-s + (−0.0717 + 0.239i)16-s + (0.0152 − 0.00553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456432 - 0.740584i\)
\(L(\frac12)\) \(\approx\) \(0.456432 - 0.740584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.893 + 0.448i)T \)
3 \( 1 + (-0.647 + 1.60i)T \)
good5 \( 1 + (0.750 + 2.50i)T + (-4.17 + 2.74i)T^{2} \)
7 \( 1 + (-0.318 - 0.738i)T + (-4.80 + 5.09i)T^{2} \)
11 \( 1 + (-0.151 + 0.0358i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (0.574 + 0.377i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (-0.0626 + 0.0228i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.221 + 0.0805i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3.45 + 8.00i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (-0.585 - 10.0i)T + (-28.8 + 3.36i)T^{2} \)
31 \( 1 + (-7.28 + 0.851i)T + (30.1 - 7.14i)T^{2} \)
37 \( 1 + (-6.12 - 5.14i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (4.49 - 2.25i)T + (24.4 - 32.8i)T^{2} \)
43 \( 1 + (-2.28 - 2.42i)T + (-2.50 + 42.9i)T^{2} \)
47 \( 1 + (-12.4 - 1.46i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (3.27 - 5.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.05 + 1.67i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (3.70 - 4.97i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (-0.484 + 8.31i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (0.683 - 3.87i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.74 + 9.87i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (15.0 + 7.56i)T + (47.1 + 63.3i)T^{2} \)
83 \( 1 + (-1.94 - 0.978i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (0.587 + 3.33i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-2.07 + 6.93i)T + (-81.0 - 53.3i)T^{2} \)
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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

−12.44121825097002199239129941929, −11.84865705491358569538257158075, −10.54747531944280915082387666334, −9.029475269607285735182311769267, −8.594942625521222645694712679662, −7.60147306876240058402267158227, −6.36815678476263204201859861054, −4.70421315745547264587977367909, −2.79919556725813851265055859874, −1.06541587930599642142530118216, 2.74361528887922098859392503470, 4.13436773128561259978791859694, 5.74509898166297266145120865565, 7.14736609278065272093537727953, 7.998875861587111387639764035229, 9.262146781904419212112159421894, 10.10665075342818736966915759419, 10.95866551637792152243733584448, 11.68977113886325378238178599289, 13.61043327753766998860667653022

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