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Properties

Label 2-3420-19.11-c1-0-31
Degree $2$
Conductor $3420$
Sign $-0.998 - 0.0595i$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + 2.72·7-s − 3.31·11-s + (−1.62 − 2.81i)13-s + (−1.17 + 2.03i)17-s + (−3.11 + 3.04i)19-s + (1.07 + 1.86i)23-s + (−0.499 − 0.866i)25-s + (−1.96 − 3.40i)29-s − 10.1·31-s + (1.36 − 2.36i)35-s − 3.68·37-s + (−0.363 + 0.629i)41-s + (1.18 − 2.05i)43-s + (−5.51 − 9.54i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + 1.03·7-s − 0.999·11-s + (−0.450 − 0.780i)13-s + (−0.285 + 0.494i)17-s + (−0.714 + 0.699i)19-s + (0.223 + 0.387i)23-s + (−0.0999 − 0.173i)25-s + (−0.365 − 0.632i)29-s − 1.83·31-s + (0.230 − 0.399i)35-s − 0.605·37-s + (−0.0568 + 0.0983i)41-s + (0.180 − 0.313i)43-s + (−0.803 − 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0595i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (3241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.998 - 0.0595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1786925502\)
\(L(\frac12)\) \(\approx\) \(0.1786925502\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (3.11 - 3.04i)T \)
good7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 + 3.31T + 11T^{2} \)
13 \( 1 + (1.62 + 2.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.17 - 2.03i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.07 - 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.96 + 3.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 + (0.363 - 0.629i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.18 + 2.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.51 + 9.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.49 + 7.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.48 - 9.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.22 - 7.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.87 - 8.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.45 + 5.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.24 + 2.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.99 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + (-4.27 - 7.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.61 - 6.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.191008525691173207255934899153, −7.68053223920915589544695681874, −6.83244899546296841606310247934, −5.52840019941502188211047303413, −5.43885704556363549033184268367, −4.44173610493307831832586192225, −3.54432763956695934461220090630, −2.32014676658495548225370387474, −1.60528742829682107574844668844, −0.04800966595639893860868517599, 1.70855537143620479836795679921, 2.40085211512600293802724905576, 3.41606695729502212779554077750, 4.70597434685288161732066482013, 4.93178038207402602987825465220, 5.97435585859749625628636248484, 6.89038216376026738504563455606, 7.45779303473010871247168188118, 8.181648529107511868150284490981, 9.014011320280952802002892620231

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