↑

Properties

Label 2-155-155.92-c1-0-7
Degree $2$
Conductor $155$
Sign $-0.0597 - 0.998i$
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  + (1.95 + 1.95i)2-s + 5.67i·4-s + (1.05 − 1.96i)5-s + (−2.22 − 2.22i)7-s + (−7.20 + 7.20i)8-s − 3i·9-s + (5.93 − 1.78i)10-s − 8.71i·14-s − 16.8·16-s + (5.87 − 5.87i)18-s + 8.66i·19-s + (11.1 + 6.01i)20-s + (−2.75 − 4.17i)25-s + (12.6 − 12.6i)28-s + 5.56·31-s + (−18.6 − 18.6i)32-s + ⋯
L(s)  = 1  + (1.38 + 1.38i)2-s + 2.83i·4-s + (0.473 − 0.880i)5-s + (−0.840 − 0.840i)7-s + (−2.54 + 2.54i)8-s − i·9-s + (1.87 − 0.563i)10-s − 2.32i·14-s − 4.21·16-s + (1.38 − 1.38i)18-s + 1.98i·19-s + (2.49 + 1.34i)20-s + (−0.550 − 0.834i)25-s + (2.38 − 2.38i)28-s + 1.00·31-s + (−3.29 − 3.29i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0597 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0597 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.0597 - 0.998i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.0597 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41327 + 1.50046i\)
\(L(\frac12)\) \(\approx\) \(1.41327 + 1.50046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.05 + 1.96i)T \)
31 \( 1 - 5.56T \)
good2 \( 1 + (-1.95 - 1.95i)T + 2iT^{2} \)
3 \( 1 + 3iT^{2} \)
7 \( 1 + (2.22 + 2.22i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (1.56 + 1.56i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 11.4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (11.5 + 11.5i)T + 67iT^{2} \)
71 \( 1 - 3.24T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (3.29 + 3.29i)T + 97iT^{2} \)
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

−13.39959168879484915267838761594, −12.54086969648377387743000287633, −12.00435654860721070436425666021, −9.945408533924873907146947712509, −8.743745622740357567681044106803, −7.64728707605115825036610313587, −6.44685992731625319831900545826, −5.80506063333937067762581957790, −4.38680246710618550421523038137, −3.45623786285977052810451848857, 2.33480982108176893494312664372, 3.04477056441044182851741971756, 4.73854685034716931125035673728, 5.81950719361955728743375797525, 6.78851527106588278945606875400, 9.185877965564368780605616584361, 10.07399795622493693179625457282, 10.95479737236240437849046258531, 11.66979588755324658315291108340, 12.87497615006618425541466836978

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