Properties

Label 2-3420-95.49-c1-0-21
Degree $2$
Conductor $3420$
Sign $0.934 + 0.355i$
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  + (−1.82 + 1.29i)5-s − 2.54i·7-s − 2.22·11-s + (−6.08 + 3.51i)13-s + (2.21 + 1.27i)17-s + (2.70 − 3.41i)19-s + (−6.95 + 4.01i)23-s + (1.65 − 4.71i)25-s + (0.941 + 1.63i)29-s + 5.98·31-s + (3.28 + 4.64i)35-s + 2.86i·37-s + (3.67 − 6.36i)41-s + (−3.19 − 1.84i)43-s + (4.09 − 2.36i)47-s + ⋯
L(s)  = 1  + (−0.816 + 0.578i)5-s − 0.961i·7-s − 0.670·11-s + (−1.68 + 0.973i)13-s + (0.537 + 0.310i)17-s + (0.620 − 0.784i)19-s + (−1.44 + 0.837i)23-s + (0.331 − 0.943i)25-s + (0.174 + 0.302i)29-s + 1.07·31-s + (0.555 + 0.784i)35-s + 0.470i·37-s + (0.573 − 0.994i)41-s + (−0.487 − 0.281i)43-s + (0.597 − 0.344i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.934 + 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041672666\)
\(L(\frac12)\) \(\approx\) \(1.041672666\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.82 - 1.29i)T \)
19 \( 1 + (-2.70 + 3.41i)T \)
good7 \( 1 + 2.54iT - 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + (6.08 - 3.51i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.21 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.95 - 4.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.941 - 1.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 - 2.86iT - 37T^{2} \)
41 \( 1 + (-3.67 + 6.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.19 + 1.84i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.09 + 2.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.91 - 5.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.17 - 7.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.13 + 7.16i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.9 - 6.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.13 - 3.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 + (-7.19 - 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.04 + 2.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.285464656248340988940261796476, −7.68785747961916545937262719349, −7.17856740034318533732436312375, −6.61600845777758686552629240742, −5.41428921500619670860386007560, −4.59067839978105406844268274301, −3.93628326242988084759780988410, −3.02384114531867683699355562403, −2.08764105155139581372616022141, −0.49987582363334691852378781752, 0.67591301679333904071987169836, 2.29348326941872996287179300854, 2.94511435571989678424214101874, 4.04055438192845047971107451070, 5.03454913080081146987340799421, 5.36602342349234061567985397296, 6.31854535182863265563119670278, 7.45785243055726957520953165493, 8.072158231037997939133249597039, 8.264942219496262190399610814833

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