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Properties

Label 2-3420-15.2-c1-0-13
Degree $2$
Conductor $3420$
Sign $0.872 - 0.487i$
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  + (1.76 + 1.37i)5-s + (−1.90 − 1.90i)7-s − 1.74i·11-s + (−2.35 + 2.35i)13-s + (−1.18 + 1.18i)17-s + i·19-s + (−6.08 − 6.08i)23-s + (1.24 + 4.84i)25-s + 9.34·29-s + 7.30·31-s + (−0.754 − 5.96i)35-s + (5.41 + 5.41i)37-s + 5.78i·41-s + (−0.257 + 0.257i)43-s + (−3.22 + 3.22i)47-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)5-s + (−0.718 − 0.718i)7-s − 0.527i·11-s + (−0.654 + 0.654i)13-s + (−0.287 + 0.287i)17-s + 0.229i·19-s + (−1.26 − 1.26i)23-s + (0.249 + 0.968i)25-s + 1.73·29-s + 1.31·31-s + (−0.127 − 1.00i)35-s + (0.890 + 0.890i)37-s + 0.903i·41-s + (−0.0392 + 0.0392i)43-s + (−0.470 + 0.470i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.872 - 0.487i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.872 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813880628\)
\(L(\frac12)\) \(\approx\) \(1.813880628\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 - 1.37i)T \)
19 \( 1 - iT \)
good7 \( 1 + (1.90 + 1.90i)T + 7iT^{2} \)
11 \( 1 + 1.74iT - 11T^{2} \)
13 \( 1 + (2.35 - 2.35i)T - 13iT^{2} \)
17 \( 1 + (1.18 - 1.18i)T - 17iT^{2} \)
23 \( 1 + (6.08 + 6.08i)T + 23iT^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 - 7.30T + 31T^{2} \)
37 \( 1 + (-5.41 - 5.41i)T + 37iT^{2} \)
41 \( 1 - 5.78iT - 41T^{2} \)
43 \( 1 + (0.257 - 0.257i)T - 43iT^{2} \)
47 \( 1 + (3.22 - 3.22i)T - 47iT^{2} \)
53 \( 1 + (-6.80 - 6.80i)T + 53iT^{2} \)
59 \( 1 - 8.76T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + (1.34 + 1.34i)T + 67iT^{2} \)
71 \( 1 - 6.99iT - 71T^{2} \)
73 \( 1 + (-8.84 + 8.84i)T - 73iT^{2} \)
79 \( 1 - 3.31iT - 79T^{2} \)
83 \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (5.34 + 5.34i)T + 97iT^{2} \)
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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

−8.564821253856442093957378437420, −8.015867281064055008281601605522, −6.77839306410175209960349489493, −6.60304610120218608966371659204, −5.90764291785553844691373369096, −4.73206225604379477336987341741, −4.03210336762736598721753490207, −2.93601716412168461755986257220, −2.30982151498448893065330493479, −0.906480153547229675160581588213, 0.68913158707574799541358704725, 2.14571962000430651312331857739, 2.67574296922264691428505564125, 3.89767719776846376433145013505, 4.91315190149121896678598757036, 5.48828546029343328278365143923, 6.23059491461197939737534728698, 6.92787033564390906215493029732, 7.931423490298519579109970450631, 8.596116320828975551639814296240

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