↑

Properties

Label 2-155-155.39-c1-0-0
Degree $2$
Conductor $155$
Sign $-0.968 + 0.247i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
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  + (1.03 + 0.334i)2-s + (−2.61 + 0.849i)3-s + (−0.669 − 0.486i)4-s + (−1.31 − 1.80i)5-s − 2.97·6-s + (−2.94 + 4.04i)7-s + (−1.79 − 2.47i)8-s + (3.68 − 2.67i)9-s + (−0.746 − 2.30i)10-s + (−1.13 − 0.821i)11-s + (2.16 + 0.702i)12-s + (0.605 − 0.196i)13-s + (−4.38 + 3.18i)14-s + (4.97 + 3.61i)15-s + (−0.513 − 1.58i)16-s + (1.41 + 1.94i)17-s + ⋯
L(s)  = 1  + (0.728 + 0.236i)2-s + (−1.50 + 0.490i)3-s + (−0.334 − 0.243i)4-s + (−0.587 − 0.809i)5-s − 1.21·6-s + (−1.11 + 1.53i)7-s + (−0.636 − 0.875i)8-s + (1.22 − 0.893i)9-s + (−0.236 − 0.728i)10-s + (−0.341 − 0.247i)11-s + (0.624 + 0.202i)12-s + (0.167 − 0.0545i)13-s + (−1.17 + 0.851i)14-s + (1.28 + 0.933i)15-s + (−0.128 − 0.395i)16-s + (0.342 + 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.968 + 0.247i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00485794 - 0.0386086i\)
\(L(\frac12)\) \(\approx\) \(0.00485794 - 0.0386086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.31 + 1.80i)T \)
31 \( 1 + (3.70 - 4.15i)T \)
good2 \( 1 + (-1.03 - 0.334i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.61 - 0.849i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (2.94 - 4.04i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.13 + 0.821i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.605 + 0.196i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.41 - 1.94i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.848 - 2.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.38 + 6.02i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.238 - 0.734i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 7.89iT - 37T^{2} \)
41 \( 1 + (2.21 - 6.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.89 + 1.59i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.22 + 1.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.788 + 1.08i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.16 + 3.57i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 + 3.34iT - 67T^{2} \)
71 \( 1 + (-1.69 + 1.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.46 - 8.89i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.88 + 4.27i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.25 - 2.68i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (8.03 + 5.83i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.1 - 13.9i)T + (-29.9 - 92.2i)T^{2} \)
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

−13.01316921236175232840392878831, −12.41700262654279495342510034500, −11.96608867316570564854007564079, −10.54160450647541127261444199767, −9.537132948338602355474082337819, −8.530170288298319438107046940599, −6.37984811877942997408444907064, −5.74068593704616748420910290446, −4.98528341125077247686498177237, −3.71703997165285702035633078243, 0.03489792840401381753712957528, 3.37319684413297616729961083200, 4.44489043375792237600304309743, 5.85861366853868500606065395365, 6.91598107414058841157475029403, 7.66438822514525754554500103916, 9.805269899176130207184062185223, 10.76488958614890340591965925902, 11.57588728906023377994014147214, 12.36589347402410465105793882797

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