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Properties

Label 2-3420-285.113-c0-0-5
Degree $2$
Conductor $3420$
Sign $0.139 + 0.990i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (−0.965 − 0.258i)5-s + (1.36 − 1.36i)7-s + 0.517i·11-s + (−1.22 − 1.22i)17-s + i·19-s + (1.41 − 1.41i)23-s + (0.866 + 0.499i)25-s + (−1.67 + 0.965i)35-s + (−1.36 − 1.36i)43-s + (0.707 + 0.707i)47-s − 2.73i·49-s + (0.133 − 0.499i)55-s − 61-s + (−0.366 − 0.366i)73-s + (0.707 + 0.707i)77-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)5-s + (1.36 − 1.36i)7-s + 0.517i·11-s + (−1.22 − 1.22i)17-s + i·19-s + (1.41 − 1.41i)23-s + (0.866 + 0.499i)25-s + (−1.67 + 0.965i)35-s + (−1.36 − 1.36i)43-s + (0.707 + 0.707i)47-s − 2.73i·49-s + (0.133 − 0.499i)55-s − 61-s + (−0.366 − 0.366i)73-s + (0.707 + 0.707i)77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.139 + 0.990i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120472176\)
\(L(\frac12)\) \(\approx\) \(1.120472176\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
23 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
47 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.491566088660749417554420444692, −7.84249112866920095583081047410, −7.13702706768030264447704128887, −6.80096464050794788626067728220, −5.26075770628720146659244228842, −4.51574228467872569931676257980, −4.31381591137200768241679156589, −3.16130538638800040075147945219, −1.86350072823849887079897126766, −0.71056888543002195965574411535, 1.47349202592697686294977616376, 2.52850199688887599139407523924, 3.39472759488715313151029699830, 4.49475485109328180764802784075, 5.03494047780526172399237654449, 5.89633086124636026734407078712, 6.77786077457702827815064967201, 7.59739987589902724409266360472, 8.361476311292058912676567152109, 8.718143770335932839838332816347

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