Properties

Label 2-322-161.111-c1-0-10
Degree $2$
Conductor $322$
Sign $0.887 + 0.460i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (0.415 + 0.909i)2-s + (−0.760 − 2.59i)3-s + (−0.654 + 0.755i)4-s + (2.71 + 1.74i)5-s + (2.04 − 1.76i)6-s + (0.0953 − 2.64i)7-s + (−0.959 − 0.281i)8-s + (−3.61 + 2.32i)9-s + (−0.459 + 3.19i)10-s + (0.903 + 0.412i)11-s + (2.45 + 1.12i)12-s + (5.14 + 0.739i)13-s + (2.44 − 1.01i)14-s + (2.45 − 8.37i)15-s + (−0.142 − 0.989i)16-s + (−4.81 − 5.55i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−0.439 − 1.49i)3-s + (−0.327 + 0.377i)4-s + (1.21 + 0.781i)5-s + (0.833 − 0.721i)6-s + (0.0360 − 0.999i)7-s + (−0.339 − 0.0996i)8-s + (−1.20 + 0.773i)9-s + (−0.145 + 1.01i)10-s + (0.272 + 0.124i)11-s + (0.709 + 0.323i)12-s + (1.42 + 0.205i)13-s + (0.653 − 0.270i)14-s + (0.634 − 2.16i)15-s + (−0.0355 − 0.247i)16-s + (−1.16 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51968 - 0.370351i\)
\(L(\frac12)\) \(\approx\) \(1.51968 - 0.370351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.0953 + 2.64i)T \)
23 \( 1 + (-3.70 + 3.04i)T \)
good3 \( 1 + (0.760 + 2.59i)T + (-2.52 + 1.62i)T^{2} \)
5 \( 1 + (-2.71 - 1.74i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (-0.903 - 0.412i)T + (7.20 + 8.31i)T^{2} \)
13 \( 1 + (-5.14 - 0.739i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (4.81 + 5.55i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-1.19 + 1.38i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-0.920 - 1.06i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (1.36 - 4.64i)T + (-26.0 - 16.7i)T^{2} \)
37 \( 1 + (-3.29 - 5.13i)T + (-15.3 + 33.6i)T^{2} \)
41 \( 1 + (5.69 - 8.86i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.81 - 9.59i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + 7.29iT - 47T^{2} \)
53 \( 1 + (6.00 - 0.863i)T + (50.8 - 14.9i)T^{2} \)
59 \( 1 + (5.52 + 0.794i)T + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (-3.21 - 0.944i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (5.13 - 2.34i)T + (43.8 - 50.6i)T^{2} \)
71 \( 1 + (-0.102 - 0.223i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (0.551 + 0.478i)T + (10.3 + 72.2i)T^{2} \)
79 \( 1 + (0.298 + 0.0428i)T + (75.7 + 22.2i)T^{2} \)
83 \( 1 + (12.8 - 8.22i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (8.66 - 2.54i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-9.11 - 5.85i)T + (40.2 + 88.2i)T^{2} \)
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

−11.51041634856892076988223160161, −10.91078937640643649623841813082, −9.623172455126126258883338032399, −8.450054188742223421548466927327, −7.18072303376117017529118687561, −6.66636624394448634860961270367, −6.18932084236990328541169374817, −4.78883397993785708946263464250, −2.90947780461759022583967609761, −1.31325495300788377842799622191, 1.83837761600329605391744435250, 3.56625426106935463648053227940, 4.60585526037893285822754115574, 5.78011767986260740514382579354, 5.91301929111976232001320158197, 8.761143599206838487519328408220, 8.987230486711991477330369881707, 9.906054718255140620029005201391, 10.80016820910542042734118576844, 11.39570304450032541812914450780

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