Properties

Label 2-1340-1340.799-c0-0-1
Degree $2$
Conductor $1340$
Sign $-0.215 + 0.976i$
Analytic cond. $0.668747$
Root an. cond. $0.817769$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)2-s + (0.118 + 0.258i)3-s + (−0.142 − 0.989i)4-s + (−0.959 + 0.281i)5-s + (0.273 + 0.0801i)6-s + (0.544 − 0.627i)7-s + (−0.841 − 0.540i)8-s + (0.601 − 0.694i)9-s + (−0.415 + 0.909i)10-s + (0.239 − 0.153i)12-s + (−0.118 − 0.822i)14-s + (−0.186 − 0.215i)15-s + (−0.959 + 0.281i)16-s + (−0.130 − 0.909i)18-s + (0.415 + 0.909i)20-s + (0.226 + 0.0666i)21-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (0.118 + 0.258i)3-s + (−0.142 − 0.989i)4-s + (−0.959 + 0.281i)5-s + (0.273 + 0.0801i)6-s + (0.544 − 0.627i)7-s + (−0.841 − 0.540i)8-s + (0.601 − 0.694i)9-s + (−0.415 + 0.909i)10-s + (0.239 − 0.153i)12-s + (−0.118 − 0.822i)14-s + (−0.186 − 0.215i)15-s + (−0.959 + 0.281i)16-s + (−0.130 − 0.909i)18-s + (0.415 + 0.909i)20-s + (0.226 + 0.0666i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358603115\)
\(L(\frac12)\) \(\approx\) \(1.358603115\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 + 0.755i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
good3 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.544 + 0.627i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.142 - 0.989i)T^{2} \)
23 \( 1 + (0.830 + 1.81i)T + (-0.654 + 0.755i)T^{2} \)
29 \( 1 - 0.830T + T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09578557000034120647426903393, −8.795505232351099316179594371111, −8.079084705278566512629153920470, −6.92317592127180251210356324629, −6.36680629416234812272717485412, −4.90301524254707044969796434371, −4.28158373587060560243694466782, −3.66316149170293349531988682035, −2.57330021242881355817236143581, −1.00060713224894365491177116672, 1.93093989914354153461804952536, 3.28351745911040116971730504814, 4.23585660997075881373976597181, 5.01420730201730327700693854308, 5.75138245774138247627837687283, 6.96390462634835734993167205840, 7.58551282341249446688284090021, 8.197225847560973627841327932879, 8.813150947452425722486634673923, 9.915594999075695486684403425903

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