Properties

Label 2-161-161.100-c1-0-13
Degree $2$
Conductor $161$
Sign $-0.952 - 0.305i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 − 1.25i)2-s + (1.76 − 2.47i)3-s + (0.502 + 2.07i)4-s + (−2.84 + 0.549i)5-s + (−5.92 + 1.73i)6-s + (−2.63 + 0.260i)7-s + (0.110 − 0.242i)8-s + (−2.03 − 5.88i)9-s + (5.24 + 2.70i)10-s + (0.552 − 0.434i)11-s + (6.00 + 2.40i)12-s + (3.04 − 1.95i)13-s + (4.53 + 2.89i)14-s + (−3.66 + 8.01i)15-s + (3.30 − 1.70i)16-s + (−3.27 − 3.11i)17-s + ⋯
L(s)  = 1  + (−1.12 − 0.888i)2-s + (1.01 − 1.42i)3-s + (0.251 + 1.03i)4-s + (−1.27 + 0.245i)5-s + (−2.41 + 0.709i)6-s + (−0.995 + 0.0982i)7-s + (0.0390 − 0.0855i)8-s + (−0.678 − 1.96i)9-s + (1.65 + 0.854i)10-s + (0.166 − 0.131i)11-s + (1.73 + 0.694i)12-s + (0.845 − 0.543i)13-s + (1.21 + 0.773i)14-s + (−0.945 + 2.06i)15-s + (0.826 − 0.426i)16-s + (−0.793 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0834494 + 0.533426i\)
\(L(\frac12)\) \(\approx\) \(0.0834494 + 0.533426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 - 0.260i)T \)
23 \( 1 + (4.72 + 0.838i)T \)
good2 \( 1 + (1.59 + 1.25i)T + (0.471 + 1.94i)T^{2} \)
3 \( 1 + (-1.76 + 2.47i)T + (-0.981 - 2.83i)T^{2} \)
5 \( 1 + (2.84 - 0.549i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-0.552 + 0.434i)T + (2.59 - 10.6i)T^{2} \)
13 \( 1 + (-3.04 + 1.95i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (3.27 + 3.11i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-2.90 + 2.76i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-8.16 + 2.39i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.658 - 0.0628i)T + (30.4 - 5.86i)T^{2} \)
37 \( 1 + (-2.87 - 8.31i)T + (-29.0 + 22.8i)T^{2} \)
41 \( 1 + (0.195 + 0.225i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (2.93 + 6.42i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + (-3.07 + 5.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.00171 - 0.0359i)T + (-52.7 - 5.03i)T^{2} \)
59 \( 1 + (-1.03 - 0.532i)T + (34.2 + 48.0i)T^{2} \)
61 \( 1 + (0.748 + 1.05i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (1.11 - 0.447i)T + (48.4 - 46.2i)T^{2} \)
71 \( 1 + (1.99 - 13.8i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (0.289 + 1.19i)T + (-64.8 + 33.4i)T^{2} \)
79 \( 1 + (0.169 + 3.54i)T + (-78.6 + 7.50i)T^{2} \)
83 \( 1 + (-7.99 + 9.22i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-11.7 - 1.12i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (1.51 + 1.74i)T + (-13.8 + 96.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.99832765774048033737459753180, −11.62889296154104776180716471263, −10.20902386788805182619384910863, −8.978602337722509480428068839138, −8.347618493089077265090184576842, −7.48658444997203497737744830024, −6.48195113586280397339360341800, −3.47459431210530590428012275045, −2.60162290782447156972204715984, −0.66506896995692757795492370831, 3.49645953302534030438001838024, 4.22225216576648467323937488592, 6.26085255965382024077056859692, 7.66540262472126346099398329415, 8.452041278237618130837779580678, 9.140982862167710014855839383747, 9.961993372211636215069096810778, 10.90334182371402359085923405590, 12.37240418504198635798309019092, 13.82017111697639109846352825367

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