↑

Properties

Label 2-342-19.16-c1-0-1
Degree $2$
Conductor $342$
Sign $0.692 - 0.721i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.0812 − 0.460i)5-s + (2.20 + 3.82i)7-s + (0.500 − 0.866i)8-s + (−0.358 + 0.300i)10-s + (−2.76 + 4.79i)11-s + (−5.62 − 2.04i)13-s + (0.766 − 4.34i)14-s + (−0.939 + 0.342i)16-s + (3.33 + 2.79i)17-s + (4.34 − 0.405i)19-s + 0.467·20-s + (5.19 − 1.89i)22-s + (0.549 + 3.11i)23-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.0363 − 0.206i)5-s + (0.833 + 1.44i)7-s + (0.176 − 0.306i)8-s + (−0.113 + 0.0951i)10-s + (−0.833 + 1.44i)11-s + (−1.55 − 0.567i)13-s + (0.204 − 1.16i)14-s + (−0.234 + 0.0855i)16-s + (0.807 + 0.677i)17-s + (0.995 − 0.0929i)19-s + 0.104·20-s + (1.10 − 0.403i)22-s + (0.114 + 0.649i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.692 - 0.721i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890498 + 0.379851i\)
\(L(\frac12)\) \(\approx\) \(0.890498 + 0.379851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 \)
19 \( 1 + (-4.34 + 0.405i)T \)
good5 \( 1 + (-0.0812 + 0.460i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.20 - 3.82i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.76 - 4.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.62 + 2.04i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.33 - 2.79i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.549 - 3.11i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.15 + 0.970i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.09 - 1.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.75T + 37T^{2} \)
41 \( 1 + (1.84 - 0.669i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.624 + 3.54i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-7.08 + 5.94i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.464 - 2.63i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.01 + 0.853i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.15 + 6.53i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.90 - 1.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.31 + 7.48i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.97 - 1.08i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.18 - 0.432i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (8.96 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.7 - 4.26i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.36 + 5.34i)T + (16.8 + 95.5i)T^{2} \)
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

−11.97647858678465013113761684261, −10.54399642371925529735086523596, −9.819060508291013685297555922161, −8.968399067727803665619293960057, −7.908296932926371187912392729981, −7.29538273535064950627383392817, −5.41770898861100149597623126160, −4.89194418656639326084985937241, −2.89203696212028323312257847745, −1.87249957561237596505139017143, 0.840638065484889849145238027212, 2.88647675183581863557812176145, 4.56950585627728649111121170119, 5.48664641253853594985843447549, 6.97375010169060788581675326203, 7.56190262498087019012090005537, 8.378891229230037063665661039929, 9.648508963039330243905540041817, 10.45701472938804700230717205841, 11.11732309455892788432049673433

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