↑

Properties

Label 2-2340-65.8-c1-0-34
Degree $2$
Conductor $2340$
Sign $-0.966 - 0.256i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  − 2.23i·5-s + (0.381 − 0.381i)11-s + (−2 − 3i)13-s + (−2.23 − 2.23i)17-s + (−0.854 + 0.854i)19-s + (−5.61 + 5.61i)23-s − 5.00·25-s − 0.763i·29-s + (−0.854 − 0.854i)31-s + 3.70i·37-s + (−8.23 − 8.23i)41-s + (2.85 − 2.85i)43-s + 8.94i·47-s + 7·49-s + (7.47 + 7.47i)53-s + ⋯
L(s)  = 1  − 0.999i·5-s + (0.115 − 0.115i)11-s + (−0.554 − 0.832i)13-s + (−0.542 − 0.542i)17-s + (−0.195 + 0.195i)19-s + (−1.17 + 1.17i)23-s − 1.00·25-s − 0.141i·29-s + (−0.153 − 0.153i)31-s + 0.609i·37-s + (−1.28 − 1.28i)41-s + (0.435 − 0.435i)43-s + 1.30i·47-s + 49-s + (1.02 + 1.02i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3508478336\)
\(L(\frac12)\) \(\approx\) \(0.3508478336\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
13 \( 1 + (2 + 3i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + (-0.381 + 0.381i)T - 11iT^{2} \)
17 \( 1 + (2.23 + 2.23i)T + 17iT^{2} \)
19 \( 1 + (0.854 - 0.854i)T - 19iT^{2} \)
23 \( 1 + (5.61 - 5.61i)T - 23iT^{2} \)
29 \( 1 + 0.763iT - 29T^{2} \)
31 \( 1 + (0.854 + 0.854i)T + 31iT^{2} \)
37 \( 1 - 3.70iT - 37T^{2} \)
41 \( 1 + (8.23 + 8.23i)T + 41iT^{2} \)
43 \( 1 + (-2.85 + 2.85i)T - 43iT^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 + (-7.47 - 7.47i)T + 53iT^{2} \)
59 \( 1 + (-5.61 - 5.61i)T + 59iT^{2} \)
61 \( 1 + 7.70T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + (0.381 + 0.381i)T + 71iT^{2} \)
73 \( 1 + 1.70T + 73T^{2} \)
79 \( 1 + 9.70iT - 79T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 + (2.23 + 2.23i)T + 89iT^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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

−8.669327387741087078574355921037, −7.78209862949692757261896013658, −7.25457220802436929328948381605, −5.98169894055157387845223966084, −5.47375002908628416892917123245, −4.56765499510918407027500004163, −3.78882815439641341704882468638, −2.59250243922632381979724608549, −1.45339196178212638740061704999, −0.11299901228356364525843913683, 1.88887270930267824671142839082, 2.61604541478301970350091897361, 3.79888700317608936464551670137, 4.45448114811542446905835524265, 5.58885791040352931087254633187, 6.59245871559636288944436375498, 6.83665747394534455887053279043, 7.85589192067355018568562349744, 8.602477566993892089824547954606, 9.457760218397812451678791671071

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